In preparation for my upcoming course on random matrices, I am briefly reviewing some relevant foundational aspects of probability theory, as well as setting up basic probabilistic notation that we will be using in later posts. This is quite basic material for a graduate course, and somewhat pedantic in nature, but given how heavily we will be relying on probability theory in this course, it seemed appropriate to take some time to go through these issues carefully.
We will certainly not attempt to cover all aspects of probability theory in this review. Aside from the utter foundations, we will be focusing primarily on those probabilistic concepts and operations that are useful for bounding the distribution of random variables, and on ensuring convergence of such variables as one sends a parameter off to infinity.
We will assume familiarity with the foundations of measure theory; see for instance these earlier lecture notes of mine for a quick review of that topic. This is also not intended to be a first introduction to probability theory, but is instead a revisiting of these topics from a graduate-level perspective (and in particular, after one has understood the foundations of measure theory). Indeed, I suspect it will be almost impossible to follow this course without already having a firm grasp of undergraduate probability theory.
— 1. Foundations —
At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate. At a practical level, the opposite is true: just as number theorists study concepts (e.g. primality) that have the same meaning in every numeral system that models the natural numbers, we shall see that probability theorists study concepts (e.g. independence) that have the same meaning in every measure space that models a family of events or random variables. And indeed, just as the natural numbers can be defined abstractly without reference to any numeral system (e.g. by the Peano axioms), core concepts of probability theory, such as random variables, can also be defined abstractly, without explicit mention of a measure space; we will return to this point when we discuss free probability later in this course.
For now, though, we shall stick to the standard measure-theoretic approach to probability theory. In this approach, we assume the presence of an ambient sample space , which intuitively is supposed to describe all the possible outcomes of all the sources of randomness that one is studying. Mathematically, this sample space is a probability space – a set , together with a -algebra of subsets of (the elements of which we will identify with the probabilistic concept of an event), and a probability measure on the space of events, i.e. an assignment of a real number in to every event (known as the probability of that event), such that the whole space has probability , and such that is countably additive.
Elements of the sample space will be denoted . However, for reasons that will be explained shortly, we will try to avoid actually referring to such elements unless absolutely required to.
If we were studying just a single random process, e.g. rolling a single die, then one could choose a very simple sample space – in this case, one could choose the finite set , with the discrete -algebra and the uniform probability measure. But if one later wanted to also study additional random processes (e.g. supposing one later wanted to roll a second die, and then add the two resulting rolls), one would have to change the sample space (e.g. to change it now to the product space ). If one was particularly well organised, one could in principle work out in advance all of the random variables one would ever want or need, and then specify the sample space accordingly, before doing any actual probability theory. In practice, though, it is far more convenient to add new sources of randomness on the fly, if and when they are needed, and extend the sample space as necessary. This point is often glossed over in introductory probability texts, so let us spend a little time on it. We say that one probability space extends another if there is a surjective map which is measurable (i.e. for every ) and probability preserving (i.e. for every ). (Strictly speaking, it is the pair which is the extension of , not just the space , but let us abuse notation slightly here.) By definition, every event in the original probability space is canonically identified with an event of the same probability in the extension.
Example 1 As mentioned earlier, the sample space , that models the roll of a single die, can be extended to the sample space that models the roll of the original die together with a new die, with the projection map being given by .
Another example of an extension map is that of a permutation – for instance, replacing the sample space by the isomorphic space by mapping to , etc. This extension is not actually adding any new sources of randomness, but is merely reorganising the existing randomness present in the sample space.
In order to have the freedom to perform extensions every time we need to introduce a new source of randomness, we will try to adhere to the following important dogma: probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space. (This is analogous to how differential geometry is only “allowed” to study concepts and perform operations that are preserved with respect to coordinate change, or how graph theory is only “allowed” to study concepts and perform operations that are preserved with respect to relabeling of the vertices, etc..) As long as one is adhering strictly to this dogma, one can insert as many new sources of randomness (or reorganise existing sources of randomness) as one pleases; but if one deviates from this dogma and uses specific properties of a single sample space, then one has left the category of probability theory and must now take care when doing any subsequent operation that could alter that sample space. This dogma is an important aspect of the probabilistic way of thinking, much as the insistence on studying concepts and performing operations that are invariant with respect to coordinate changes or other symmetries is an important aspect of the modern geometric way of thinking. With this probabilistic viewpoint, we shall soon see the sample space essentially disappear from view altogether, after a few foundational issues are dispensed with.
Let’s give some simple examples of what is and what is not a probabilistic concept or operation. The probability of an event is a probabilistic concept; it is preserved under extensions. Similarly, boolean operations on events such as union, intersection, and complement are also preserved under extensions and are thus also probabilistic operations. The emptiness or non-emptiness of an event is also probabilistic, as is the equality or non-equality of two events (note how it was important here that we demanded the map to be surjective in the definition of an extension). On the other hand, the cardinality of an event is not a probabilistic concept; for instance, the event that the roll of a given die gives has cardinality one in the sample space , but has cardinality six in the sample space when the values of an additional die are used to extend the sample space. Thus, in the probabilistic way of thinking, one should avoid thinking about events as having cardinality, except to the extent that they are either empty or non-empty.
Indeed, once one is no longer working at the foundational level, it is best to try to suppress the fact that events are being modeled as sets altogether. To assist in this, we will choose notation that avoids explicit use of set theoretic notation. For instance, the union of two events will be denoted rather than , and will often be referred to by a phrase such as “the event that at least one of or holds”. Similarly, the intersection will instead be denoted , or “the event that and both hold”, and the complement will instead be denoted , or “the event that does not hold” or “the event that fails”. In particular the sure event can now be referred to without any explicit mention of the sample space as . We will continue to use the subset notation (since the notation may cause confusion), but refer to this statement as “ is contained in ” or “ implies ” or “ holds only if holds” rather than “ is a subset of “, again to downplay the role of set theory in modeling these events.
We record the trivial but fundamental union bound
for any finite or countably infinite collection of events . Taking complements, we see that if each event fails with probability at most , then the joint event fails with probability at most . Thus, if one wants to ensure that all the events hold at once with a reasonable probability, one can try to do this by showing that the failure rate of the individual is small compared to the number of events one is controlling. This is a reasonably efficient strategy so long as one expects the events to be genuinely “different” from each other; if there are plenty of repetitions, then the union bound is poor (consider for instance the extreme case when does not even depend on ).
We will sometimes refer to use of the union bound to bound probabilities as the zeroth moment method, to contrast it with the first moment method, second moment method, exponential moment method, and Fourier moment methods for bounding probabilities that we will encounter later in this course.
Let us formalise some specific cases of the union bound that we will use frequently in the course. In most of this course, there will be an integer parameter , which will often be going off to infinity, and upon which most other quantities will depend; for instance, we will often be considering the spectral properties of random matrices.
Definition 1 (Asymptotic notation) We use , , , or to denote the estimate for some independent of and all . If we need to depend on a parameter, e.g. , we will indicate this by subscripts, e.g. . We write if for some that goes to zero as . We write or if .
Given an event depending on such a parameter , we have five notions (in decreasing order of confidence) that an event is likely to hold:
- An event holds surely (or is true) if it is equal to the sure event .
- An event holds almost surely (or with full probability) if it occurs with probability : .
- An event holds with overwhelming probability if, for every fixed , it holds with probability (i.e. one has for some independent of ).
- An event holds with high probability if it holds with probability for some independent of (i.e. one has for some independent of ).
- An event holds asymptotically almost surely if it holds with probability , thus the probability of success goes to in the limit .
Of course, all of these notions are probabilistic notions.
Given a family of events depending on some parameter , we say that each event in the family holds with overwhelming probability uniformly in if the constant in the definition of overwhelming probability is independent of ; one can similarly define uniformity in the concepts of holding with high probability or asymptotic almost sure probability.
From the union bound (1) we immediately have
- If is an arbitrary family of events that each hold surely, then holds surely.
- If is an at most countable family of events that each hold almost surely, then holds almost surely.
- If is a family of events of polynomial cardinality (i.e. cardinality ) which hold with uniformly overwhelming probability, the holds with overwhelming probability.
- If is a family of events of sub-polynomial cardinality (i.e. cardinality ) which hold with uniformly high probability, the holds with high probability. (In particular, the cardinality can be polylogarithmic in size, .)
- If is a family of events of uniformly bounded cardinality (i.e. cardinality ) which each hold asymptotically almost surely, then holds asymptotically almost surely. (Note that uniformity of asymptotic almost sureness is automatic when the cardinality is bounded.)
Note how as the certainty of an event gets stronger, the number of times one can apply the union bound increases. In particular, holding with overwhelming probability is practically as good as holding surely or almost surely in many of our applications (except when one has to deal with the entropy of an -dimensional system, which can be exponentially large, and will thus require a certain amount of caution).
— 2. Random variables —
An event can be in just one of two states: the event can hold or fail, with some probability assigned to each. But we will usually need to consider the more general class of random variables which can be in multiple states.
Definition 3 (Random variable) Let be a measurable space (i.e. a set , equipped with a -algebra of subsets of ). A random variable taking values in (or an -valued random variable) is a measurable map from the sample space to , i.e. a function such that is an event for every .
As the notion of a random variable involves the sample space, one has to pause to check that it invariant under extensions before one can assert that it is a probabilistic concept. But this is clear: if is a random variable, and is an extension of , then is also a random variable, which generates the same events in the sense that for every .
At this point let us make the convenient convention (which we have in fact been implicitly using already) that an event is identified with the predicate which is true on the event set and false outside of the event set. Thus for instance the event could be identified with the predicate ““; this is preferable to the set-theoretic notation , as it does not require explicit reference to the sample space and is thus more obviously a probabilistic notion. We will often omit the quotes when it is safe to do so, for instance is shorthand for .
Remark 1 On occasion, we will have to deal with almost surely defined random variables, which are only defined on a subset of of full probability. However, much as measure theory and integration theory is largely unaffected by modification on sets of measure zero, many probabilistic concepts, in particular probability, distribution, and expectation, are similarly unaffected by modification on events of probability zero. Thus, a lack of definedness on an event of probability zero will usually not cause difficulty, so long as there are at most countably many such events in which one of the probabilistic objects being studied is undefined. In such cases, one can usually resolve such issues by setting a random variable to some arbitrary value (e.g. ) whenever it would otherwise be undefined.
We observe a few key subclasses and examples of random variables:
- Discrete random variables, in which is the discrete -algebra, and is at most countable. Typical examples of include a countable subset of the reals or complexes, such as the natural numbers or integers. If , we say that the random variable is Boolean, while if is just a singleton set we say that the random variable is deterministic, and (by abuse of notation) we identify this random variable with itself. Note that a Boolean random variable is nothing more than an indicator function of an event , where is the event that the boolean function equals .
- Real-valued random variables, in which is the real line and is the Borel -algebra, generated by the open sets of . Thus for any real-valued random variable and any interval , we have the events ““. In particular, we have the upper tail event “” and lower tail event “” for any threshold . (We also consider the events “” and “” to be tail events; in practice, there is very little distinction between the two.)
- Complex random variables, whose range is the complex plane with the Borel -algebra. A typical event associated to a complex random variable is the small ball event “” for some complex number and some (small) radius . We refer to real and complex random variables collectively as scalar random variables.
- Given a -valued random variable , and a measurable map , the -valued random variable is indeed a random variable, and the operation of converting to is preserved under extension of the sample space and is thus probabilistic. This variable can also be defined without reference to the sample space as the unique random variable for which the identity
holds for all -measurable sets .
- Given two random variables and taking values in respectively, one can form the joint random variable with range with the product -algebra, by setting for every . One easily verifies that this is indeed a random variable, and that the operation of taking a joint random variable is a probabilistic operation. This variable can also be defined without reference to the sample space as the unique random variable for which one has and , where and are the usual projection maps from to respectively. One can similarly define the joint random variable for any family of random variables in various ranges (note here that the set of labels can be infinite or even uncountable).
- Combining the previous two constructions, given any measurable binary operation and random variables taking values in respectively, one can form the -valued random variable , and this is a probabilistic operation. Thus for instance one can add or multiply together scalar random variables, and similarly for the matrix-valued random variables that we will consider shortly. Similarly for ternary and higher order operations. A technical issue: if one wants to perform an operation (such as division of two scalar random variables) which is not defined everywhere (e.g. division when the denominator is zero). In such cases, one has to adjoin an additional “undefined” symbol to the output range . In practice, this will not be a problem as long as all random variables concerned are defined (i.e. avoid ) almost surely.
- Vector-valued random variables, which take values in a finite-dimensional vector space such as or with the Borel -algebra. One can view a vector-valued random variable as the joint random variable of its scalar component random variables .
- Matrix-valued random variables or random matrices, which take values in a space or of real or complex-valued matrices, again with the Borel -algebra, where are integers (usually we will focus on the square case ). Note here that the shape of the matrix is deterministic; we will not consider in this course matrices whose shapes are themselves random variables. One can view a matrix-valued random variable as the joint random variable of its scalar components . One can apply all the usual matrix operations (e.g. sum, product, determinant, trace, inverse, etc.) on random matrices to get a random variable with the appropriate range, though in some cases (e.g with inverse) one has to adjoin the undefined symbol as mentioned earlier.
- Point processes, which take values in the space of subsets of a space (or more precisely, on the space of multisets of , or even more precisely still as integer-valued locally finite measures on ), with the -algebra being generated by the counting functions for all precompact measurable sets . Thus, if is a point process in , and is a precompact measurable set, then the counting function is a discrete random variable in . For us, the key example of a point process comes from taking the spectrum of eigenvalues (counting multiplicity) of a random matrix . I discuss point processes further in this previous blog post. We will return to point processes (and define them more formally) later in this course.
Remark 2 A pedantic point: strictly speaking, one has to include the range of a random variable as part of that variable (thus one should really be referring to the pair rather than ). This leads to the annoying conclusion that, technically, boolean random variables are not integer-valued, integer-valued random variables are not real-valued, and real-valued random variables are not complex-valued. To avoid this issue we shall abuse notation very slightly and identify any random variable to any coextension of that random variable to a larger range space (assuming of course that the -algebras are compatible). Thus, for instance, a real-valued random variable which happens to only take a countable number of values will now be considered a discrete random variable also.
Given a random variable taking values in some range , we define the distribution of to be the probability measure on the measurable space defined by the formula
thus is the pushforward of the sample space probability measure by . This is easily seen to be a probability measure, and is also a probabilistic concept. The probability measure is also known as the law for .
We write for ; we also abuse notation slightly by writing .
We have seen that every random variable generates a probability distribution . The converse is also true:
Lemma 4 (Creating a random variable with a specified distribution) Let be a probability measure on a measurable space . Then (after extending the sample space if necessary) there exists an -valued random variable with distribution .
Proof: Extend to by using the obvious projection map from back to , and extending the probability measure on to the product measure on . The random variable then has distribution .
In the case of discrete random variables, is the discrete probability measure
where are non-negative real numbers that add up to . To put it another way, the distribution of a discrete random variable can be expressed as the sum of Dirac masses:
We list some important examples of discrete distributions:
- Dirac distributions , in which for and otherwise;
- discrete uniform distributions, in which is finite and for all ;
- (Unsigned) Bernoulli distributions, in which , , and for some parameter ;
- The signed Bernoulli distribution, in which and ;
- Lazy signed Bernoulli distributions, in which , , and for some parameter ;
- Geometric distributions, in which and for all natural numbers and some parameter ; and
- Poisson distributions, in which and for all natural numbers and some parameter .
Now we turn to non-discrete random variables taking values in some range . We say that a random variable is continuous if for all (here we assume that all points are measurable). If is already equipped with some reference measure (e.g. Lebesgue measure in the case of scalar, vector, or matrix-valued random variables), we say that the random variable is absolutely continuous if for all null sets in . By the Radon-Nikodym theorem, we can thus find a non-negative, absolutely integrable function with such that
for all measurable sets . More succinctly, one has
We call the probability density function of the probability distribution (and thus, of the random variable ). As usual in measure theory, this function is only defined up to almost everywhere equivalence, but this will not cause any difficulties.
In the case of real-valued random variables , the distribution can also be described in terms of the cumulative distribution function
Indeed, is the Lebesgue-Stieltjes measure of , and (in the absolutely continuous case) the derivative of exists and is equal to the probability density function almost everywhere. We will not use the cumulative distribution function much in this course, although we will be very interested in bounding tail events such as or .
We give some basic examples of absolutely continuous scalar distributions:
- uniform distributions, in which for some subset of the reals or complexes of finite non-zero measure, e.g. an interval in the real line, or a disk in the complex plane.
- The real normal distribution of mean and variance , given by the density function for . We isolate in particular the standard (real) normal distribution . Random variables with normal distributions are known as gaussian random variables.
- The complex normal distribution of mean and variance , given by the density function . Again, we isolate the standard complex normal distribution .
Later on, we will encounter several more scalar distributions of relevance to random matrix theory, such as the semicircular law or Marcenko-Pastur law. We will also of course encounter many matrix distributions (also known as matrix ensembles) as well as point processes.
Given an unsigned random variable (i.e. a random variable taking values in ), one can define the expectation or mean as the unsigned integral
which by the Fubini-Tonelli theorem can also be rewritten as
The expectation of an unsigned variable lies in also . If is a scalar random variable (which is allowed to take the value ) for which , we say that is absolutely integrable, in which case we can define its expectation as
in the complex case. Similarly for vector-valued random variables (note that in finite dimensions, all norms are equivalent, so the precise choice of norm used to define is not relevant here). If is a vector-valued random variable, then is absolutely integrable if and only if the components are all absolutely integrable, in which case one has .
A deterministic scalar random variable is its own mean. An indicator function has mean . An unsigned Bernoulli variable (as defined previously) has mean , while a signed or lazy signed Bernoulli variable has mean . A real or complex gaussian variable with distribution has mean . A Poisson random variable has mean ; a geometric random variable has mean . A uniformly distributed variable on an interval has mean .
A fundamentally important property of expectation is that it is linear: if are absolutely integrable scalar random variables and are finite scalars, then is also absolutely integrable and
By the Fubini-Tonelli theorem, the same result also applies to infinite sums provided that is finite.
We will use linearity of expectation so frequently in the sequel that we will often omit an explicit reference to it when it is being used. It is important to note that linearity of expectation requires no assumptions of independence or dependence amongst the individual random variables ; this is what makes this property of expectation so powerful.
In the unsigned (or real absolutely integrable) case, expectation is also monotone: if is true for some unsigned or real absolutely integrable , then . Again, we will usually use this basic property without explicit mentioning it in the sequel.
For an unsigned random variable, we have the obvious but very useful Markov inequality
for any , as can be seen by taking expectations of the inequality . For signed random variables, Markov’s inequality becomes
Another fact related to Markov’s inequality is that if is an unsigned or real absolutely integrable random variable, then must hold with positive probability, and also must also hold with positive probability. Use of these facts or (13), (14), combined with monotonicity and linearity of expectation, is collectively referred to as the first moment method. This method tends to be particularly easy to use (as one does not need to understand dependence or independence), but by the same token often gives sub-optimal results (as one is not exploiting any independence in the system).
Exercise 1 (Borel-Cantelli lemma) Let be a sequence of events such that . Show that almost surely, at most finitely many of the events occur at once. State and prove a result to the effect that the condition cannot be weakened.
If is an absolutely integrable or unsigned scalar random variable, and is a measurable function from the scalars to the unsigned extended reals , then one has the change of variables formula
when is complex-valued. The same formula applies to signed or complex if it is known that is absolutely integrable. Important examples of expressions such as are moments
for various (particularly ), exponential moments
for real , , and Fourier moments (or the characteristic function)
for complex or vector-valued , where denotes a real inner product. We shall also occasionally encounter the resolvents
for complex , though one has to be careful now with the absolute convergence of this random variable. Similarly, we shall also occasionally encounter negative moments of , particularly for . We also sometimes use the zeroth moment , where we take the somewhat unusual convention that for non-negative , thus for and . Thus, for instance, the union bound (1) can be rewritten (for finitely many , at least) as
for any scalar random variables and scalars (compare with (12)).
It will be important to know if a scalar random variable is “usually bounded”. We have several ways of quantifying this, in decreasing order of strength:
- is surely bounded if there exists an such that surely.
- is almost surely bounded if there exists an such that almost surely.
- is subgaussian if there exist such that for all .
- has sub-exponential tail if there exist such that for all .
- has finite moment for some if there exists such that .
- is absolutely integrable if .
- is almost surely finite if almost surely.
Exercise 2 Show that these properties genuinely are in decreasing order of strength, i.e. that each property on the list implies the next.
Exercise 3 Show that each of these properties are closed under vector space operations, thus for instance if have sub-exponential tail, show that and also have sub-exponential tail for any scalar .
The various species of Bernoulli random variable are surely bounded, and any random variable which is uniformly distributed in a bounded set is almost surely bounded. Gaussians and Poisson distributions are subgaussian, while the geometric distribution merely has sub-exponential tail. Cauchy distributions are typical examples of heavy-tailed distributions which are almost surely finite, but do not have all moments finite (indeed, the Cauchy distribution does not even have finite first moment).
If we have a family of scalar random variables depending on a parameter , we say that the are uniformly surely bounded (resp. uniformly almost surely bounded, uniformly subgaussian, have uniform sub-exponential tails, or uniformly bounded moment) if the relevant parameters in the above definitions can be chosen to be independent of .
Fix . If has finite moment, say , then from Markov’s inequality (14) one has
thus we see that the higher the moments that we control, the faster the tail decay is. From the dominated convergence theorem we also have the variant
However, this result is qualitative or ineffective rather than quantitative because it provides no rate of convergence of to zero. Indeed, it is easy to construct a family of random variables of uniformly bounded moment, but for which the quantities do not converge uniformly to zero (e.g. take to be times the indicator of an event of probability for ). Because of this issue, we will often have to strengthen the property of having a uniformly bounded moment, to that of obtaining a uniformly quantitative control on the decay in (24) for a family of random variables; we will see examples of this in later lectures. However, this technicality does not arise in the important model case of identically distributed random variables, since in this case we trivially have uniformity in the decay rate of (24).
We observe some consequences of (23):
Lemma 5 Let be a scalar random variable depending on a parameter .
- If has uniformly bounded expectation, then for any independent of , we have with high probability.
- If has uniformly bounded moment, then for any , we have with probability .
- If has uniform sub-exponential tails, then we have with overwhelming probability.
Exercise 4 Show that a real-valued random variable is subgaussian if and only if there exist such that for all real , and if and only if there exists such that for all .
Exercise 5 Show that a real-valued random variable has subexponential tails if and only if there exist such that for all positive integers .
Once the second moment of a scalar random variable is finite, one can define the variance
From Markov’s inequality we thus have Chebyshev’s inequality
Upper bounds on for large are known as large deviation inequality. Chebyshev’s inequality gives a simple but still useful large deviation inequality, which becomes useful once exceeds the standard deviation of the random variable. The use of Chebyshev’s inequality, combined with a computation of variances, is known as the second moment method.
Exercise 6 (Scaling of mean and variance) If is a scalar random variable of finite mean and variance, and are scalars, show that and . In particular, if has non-zero variance, then there exist scalars such that has mean zero and variance one.
Exercise 7 We say that a real number is a median of a real-valued random variable if .
- Show that a median always exists, and if is absolutely continuous with strictly positive density function, then the median is unique.
- If has finite second moment, show that for any median .
If is subgaussian (or has sub-exponential tails with exponent ), then from dominated convergence we have the Taylor expansion
for any real or complex , thus relating the exponential and Fourier moments with the moments.
— 3. Independence —
When studying the behaviour of a single random variable , the distribution captures all the probabilistic information one wants to know about . The following exercise is one way of making this statement rigorous:
Exercise 8 Let , be random variables (on sample spaces respectively) taking values in a range , such that . Show that after extending the spaces , the two random variables are isomorphic, in the sense that there exists a probability space isomorphism (i.e. an invertible extension map whose inverse is also an extension map) such that .
However, once one studies families of random variables taking values in measurable spaces (on a single sample space ), the distribution of the individual variables are no longer sufficient to describe all the probabilistic statistics of interest; the joint distribution of the variables (i.e. the distribution of the tuple , which can be viewed as a single random variable taking values in the product measurable space ) also becomes relevant.
Example 2 Let be drawn uniformly at random from the set . Then the random variables , , and all individually have the same distribution, namely the signed Bernoulli distribution. However the pairs , , and all have different joint distributions: the first pair, by definition, is uniformly distributed in , while the second pair is uniformly distributed in , and the third pair is uniformly distributed in . Thus, for instance, if one is told that are two random variables with the Bernoulli distribution, and asked to compute the probability that , there is insufficient information to solve the problem; if were distributed as , then the probability would be , while if were distributed as , the probability would be , and if were distributed as , the probability would be . Thus one sees that one needs the joint distribution, and not just the individual distributions, to obtain a unique answer to the question.
There is however an important special class of families of random variables in which the joint distribution is determined by the individual distributions.
Definition 6 (Joint independence) A family of random variables (which may be finite, countably infinite, or uncountably infinite) is said to be jointly independent if the distribution of is the product measure of the distribution of the individual .
A family is said to be pairwise independent if the pairs are jointly independent for all distinct . More generally, is said to be -wise independent if are jointly independent for all and all distinct .
We also say that is independent of if are jointly independent.
A family of events is said to be jointly independent if their indicators are jointly independent. Similarly for pairwise independence and -wise independence.
From the theory of product measure, we have the following equivalent formulation of joint independence:
Exercise 9 Let be a family of random variables, with each taking values in a measurable space .
- Show that the are jointly independent if and only if for every collection of distinct elements of , and all measurable subsets for , one has
- Show that the necessary and sufficient condition being -wise independent is the same, except that is constrained to be at most .
In particular, a finite family of random variables , taking values in measurable spaces are jointly independent if and only if
for all measurable .
If the are discrete random variables, one can take the to be singleton sets in the above discussion.
From the above exercise we see that joint independence implies -wise independence for any , and that joint independence is preserved under permuting, relabeling, or eliminating some or all of the . A single random variable is automatically jointly independent, and so -wise independence is vacuously true; pairwise independence is the first nontrivial notion of independence in this hierarchy.
Example 3 Let be the field of two elements, let be the subspace of triples with , and let be drawn uniformly at random from . Then are pairwise independent, but not jointly independent. In particular, is independent of each of separately, but is not independent of .
Exercise 10 This exercise generalises the above example. Let be a finite field, and let be a subspace of for some finite . Let be drawn uniformly at random from . Suppose that is not contained in any coordinate hyperplane in .
- Show that each , is uniformly distributed in .
- Show that for any , that is -wise independent if and only if is not contained in any hyperplane which is definable using at most of the coordinate variables.
- Show that is jointly independent if and only if .
Informally, we thus see that imposing constraints between variables at a time can destroy -wise independence, while leaving lower-order independence unaffected.
Exercise 11 Let be the subspace of triples with , and let be drawn uniformly at random from . Then is independent of (and in particular, is independent of and separately), but are not independent of each other.
Exercise 12 We say that one random variable (with values in ) is determined by another random variable (with values in ) if there exists a (deterministic) function such that is surely true (i.e. for all ). Show that if is a family of jointly independent random variables, and is a family such that each is determined by some subfamily of the , with the disjoint as varies, then the are jointly independent also.
Exercise 13 (Determinism vs. independence) Let be random variables. Show that is deterministic if and only if it is simultaneously determined by , and independent of .
Exercise 14 Show that a complex random variable is a complex gaussian random variable (i.e. its distribution is a complex normal distribution) if and only if its real and imaginary parts are independent real gaussian random variables with the same variance. In particular, the variance of and will be half of variance of .
One key advantage of working with jointly independent random variables and events is that one can compute various probabilistic quantities quite easily. We give some key examples below.
Exercise 15 If are jointly independent events, show that
Show that the converse statement (i.e. that (28) and (29) imply joint independence) is true for , but fails for higher . Can one find a correct replacement for this converse for higher ?
- If are jointly independent random variables taking values in , show that
- If are jointly independent absolutely integrable scalar random variables, show that is absolutely integrable, and
Remark 3 The above exercise combines well with Exercise 12. For instance, if are jointly independent subgaussian variables, then from Exercises 12, 16 we see that
for any complex . This identity is a key component of the exponential moment method, which we will discuss in the next set of notes.
The following result is a key component of the second moment method.
Exercise 17 (Pairwise independence implies linearity of variance) If are pairwise independent scalar random variables of finite mean and variance, show that
and more generally
The product measure construction allows us to extend Lemma 4:
Exercise 18 (Creation of new, independent random variables) Let be a family of random variables (not necessarily independent or finite), and let be a collection (not necessarily finite) of probability measures on measurable spaces . Then, after extending the sample space if necessary, one can find a family of independent random variables, such that each has distribution , and the two families and are independent of each other.
We isolate the important case when is independent of . We say that a family of random variables is independently and identically distributed, or iid for short, if they are jointly independent and all the have the same distribution.
Corollary 7 Let be a family of random variables (not necessarily independent or finite), let be a probability measure on a measurable space , and let be an arbitrary set. Then, after extending the sample space if necessary, one can find an iid family with distribution which is independent of .
Thus, for instance, one can create arbitrarily large iid families of Bernoulli random variables, Gaussian random variables, etc., regardless of what other random variables are already in play. We thus see that the freedom to extend the underyling sample space allows us access to an unlimited source of randomness. This is in contrast to a situation studied in complexity theory and computer science, in which one does not assume that the sample space can be extended at will, and the amount of randomness one can use is therefore limited.
Remark 4 Given two probability measures on two measurable spaces , a joining or coupling of the these measures is a random variable taking values in the product space , whose individual components have distribution respectively. Exercise 18 shows that one can always couple two distributions together in an independent manner; but one can certainly create non-independent couplings as well. The study of couplings (or joinings) is particularly important in ergodic theory, but this will not be the focus of this course.
— 4. Conditioning —
Random variables are inherently non-deterministic in nature, and as such one has to be careful when applying deterministic laws of reasoning to such variables. For instance, consider the law of the excluded middle: a statement is either true or false, but not both. If this statement is a random variable, rather than deterministic, then instead it is true with some probability and false with some complementary probability . Also, applying set-theoretic constructions with random inputs can lead to sets, spaces, and other structures which are themselves random variables, which can be quite confusing and require a certain amount of technical care; consider, for instance, the task of rigorously defining a Euclidean space when the dimension is itself a random variable.
Now, one can always eliminate these difficulties by explicitly working with points in the underlying sample space , and replacing every random variable by its evaluation at that point; this removes all the randomness from consideration, making everything deterministic (for fixed ). This approach is rigorous, but goes against the “probabilistic way of thinking”, as one now needs to take some care in extending the sample space.
However, if instead one only seeks to remove a partial amount of randomness from consideration, then one can do this in a manner consistent with the probabilistic way of thinking, by introducing the machinery of conditioning. By conditioning an event to be true or false, or conditioning a random variable to be fixed, one can turn that random event or variable into a deterministic one, while preserving the random nature of other events and variables (particularly those which are independent of the event or variable being conditioned upon).
We begin by considering the simpler situation of conditioning on an event.
Definition 8 (Conditioning on an event) Let be an event (or statement) which holds with positive probability . By conditioning on the event , we mean the act of replacing the underlying sample space with the subset of where holds, and replacing the underlying probability measure by the conditional probability measure , defined by the formula
All events on the original sample space can thus be viewed as events on the conditioned space, which we model set-theoretically as the set of all in obeying . Note that this notation is compatible with (31).
All random variables on the original sample space can also be viewed as random variables on the conditioned space, by restriction. We will refer to this conditioned random variable as , and thus define conditional distribution and conditional expectation (if is scalar) accordingly.
One can also condition on the complementary event , provided that this event holds with positive probility also.
By undoing this conditioning, we revert the underlying sample space and measure back to their original (or unconditional) values. Note that any random variable which has been defined both after conditioning on , and conditioning on , can still be viewed as a combined random variable after undoing the conditioning.
Conditioning affects the underlying probability space in a manner which is different from extension, and so the act of conditioning is not guaranteed to preserve probabilistic concepts such as distribution, probability, or expectation. Nevertheless, the conditioned version of these concepts are closely related to their unconditional counterparts:
Exercise 19 If and both occur with positive probability, establish the identities
for any (unconditional) event and
for any (unconditional) random variable (in the original sample space). In a similar spirit, if is a non-negative or absolutely integrable scalar (unconditional) random variable, show that , are also non-negative and absolutely integrable on their respective conditioned spaces, and that
In the degenerate case when occurs with full probability, conditioning to the complementary event is not well defined, but show that in those cases we can still obtain the above formulae if we adopt the convention that any term involving the vanishing factor should be omitted. Similarly if occurs with zero probability.
The above identities allow one to study probabilities, distributions, and expectations on the original sample space by conditioning to the two conditioned spaces.
From (32) we obtain the inequality
thus conditioning can magnify probabilities by a factor of at most . In particular,
- If occurs unconditionally surely, it occurs surely conditioning on also.
- If occurs unconditionally almost surely, it occurs almost surely conditioning on also.
- If occurs unconditionally with overwhelming probability, it occurs with overwhelming probability conditioning on also, provided that for some independent of .
- If occurs unconditionally with high probability, it occurs with high probability conditioning on also, provided that for some and some sufficiently small independent of .
- If occurs unconditionally asymptotically almost surely, it occurs asymptotically almost surely conditioning on also, provided that for some independent of .
Conditioning can distort the probability of events and the distribution of random variables. Most obviously, conditioning on elevates the probability of to , and sends the probability of the complementary event to zero. In a similar spirit, if is a random variable uniformly distributed on some finite set , and is a non-empty subset of , then conditioning to the event alters the distribution of to now become the uniform distribution on rather than (and conditioning to the complementary event produces the uniform distribution on ).
However, events and random variables that are independent of the event being conditioned upon are essentially unaffected by conditioning. Indeed, if is an event independent of , then occurs with the same probability as ; and if is a random variable independent of (or equivalently, independently of the indicator ), then has the same distribution as .
Remark 5 One can view conditioning to an event and its complement as the probabilistic analogue of the law of the excluded middle. In deterministic logic, given a statement , one can divide into two separate cases, depending on whether is true or false; and any other statement is unconditionally true if and only if it is conditionally true in both of these two cases. Similarly, in probability theory, given an event , one can condition into two separate sample spaces, depending on whether is conditioned to be true or false; and the unconditional statistics of any random variable or event are then a weighted average of the conditional statistics on the two sample spaces, where the weights are given by the probability of and its complement.
Now we consider conditioning with respect to a discrete random variable , taking values in some range . One can condition on any event , which occurs with positive probability. It is then not difficult to establish the analogous identities to those in Exercise 19:
Exercise 20 Let be a discrete random variable with range . Then we have
for any (unconditional) event , and
for any (unconditional) random variable (where the sum of non-negative measures is defined in the obvious manner), and for absolutely integrable or non-negative (unconditional) random variables , one has
In all of these identities, we adopt the convention that any term involving is ignored when .
With the notation as in the above exercise, we define the conditional probability of an (unconditional) event conditioning on to be the (unconditional) random variable that is defined to equal whenever , and similarly, for any absolutely integrable or non-negative (unconditional) random variable , we define the conditional expectation to be the (unconditional) random variable that is defined to equal whenever . (Strictly speaking, since we are not defining conditional expectation when , these random variables are only defined almost surely, rather than surely, but this will not cause difficulties in practice; see Remark 1.) Thus (36), (38) simplify to
Remark 6 One can interpret conditional expectation as a type of orthogonal projection; see for instance these previous lecture notes of mine. But we will not use this perspective in this course. Just as conditioning on an event and its complement can be viewed as the probabilistic analogue of the law of the excluded middle, conditioning on a discrete random variable can be viewed as the probabilistic analogue of dividing into finitely or countably many cases. For instance, one could condition on the outcome of a six-sided die, thus conditioning the underlying sample space into six separate subspaces. If the die is fair, then the unconditional statistics of a random variable or event would be an unweighted average of the conditional statistics of the six conditioned subspaces; if the die is weighted, one would take a weighted average instead.
Example 4 Let be iid signed Bernoulli random variables, and let , thus is a discrete random variable taking values in (with probability , , respectively). Then remains a signed Bernoulli random variable when conditioned to , but becomes the deterministic variable when conditioned to , and similarly becomes the deterministic variable when conditioned to . As a consequence, the conditional expectation is equal to when , when , and when ; thus . Similarly ; summing and using the linearity of (conditional) expectation (which follows automatically from the unconditional version) we obtain the obvious identity .
If are independent, then for all (with the convention that those for which are ignored), which implies in particular (for absolutely integrable ) that
(so in this case the conditional expectation is a deterministic quantity).
Example 5 Let be bounded scalar random variables (not necessarily independent), with discrete. Then we have
where the latter equality holds since clearly becomes deterministic after conditioning on .
We will also need to condition with respect to continuous random variables (this is the probabilistic analogue of dividing into a potentially uncountable number of cases). To do this formally, we need to proceed a little differently from the discrete case, introducing the notion of a disintegration of the underlying sample space.
Definition 9 (Disintegration) Let be a random variable with range . A disintegration of the underlying sample space with respect to is a subset of of full measure in (thus almost surely), together with assignment of a probability measure on the subspace of for each , which is measurable in the sense that the map is measurable for every event , and such that
for all such events, where is the (almost surely defined) random variable defined to equal whenever .
Given such a disintegration, we can then condition to the event for any by replacing with the subspace (with the induced -algebra), but replacing the underlying probability measure with . We can thus condition (unconditional) events and random variables to this event to create conditioned events and random variables on the conditioned space, giving rise to conditional probabilities (which is consistent with the existing notation for this expression) and conditional expectations (assuming absolute integrability in this conditioned space). We then set to be the (almost surely defined) random variable defined to equal whenever .
Example 6 (Discrete case) If is a discrete random variable, one can set to be the essential range of , which in the discrete case is the set of all for which . For each , we define to be the conditional probability measure relative to the event , as defined in Definition 8. It is easy to verify that this is indeed a disintegration; thus the continuous notion of conditional probability generalises the discrete one.
Example 7 (Independent case) Starting with an initial sample space , and a probability measure on a measurable space , one can adjoin a random variable taking values in with distribution that is independent of all previously existing random variables, by extending to as in Lemma 4. One can then disintegrate by taking and letting be the probability measure on induced by the obvious isomorphism between and ; this is easily seen to be a disintegration. Note that if is any random variable from the original space , then has the same distribution as for any .
Example 8 Let with Lebesgue measure, and let be the coordinate random variables of , thus are iid with the uniform distribution on . Let be the random variable with range . Then one can disintegrate by taking and letting be normalised Lebesgue measure on the diagonal line segment .
Exercise 21 (Almost uniqueness of disintegrations) Let , be two disintegrations of the same random variable . Show that for any event , one has for -almost every , where the conditional probabilities and are defined using the disintegrations , respectively. (Hint: argue by contradiction, and consider the set of for which exceeds (or vice versa) by some fixed .)
Similarly, for a scalar random variable , show that for -almost every , that is absolutely integrable with respect to the first disintegration if and only if it is absolutely integrable with respect to the second integration, and one has in such cases.
Remark 7 Under some mild topological assumptions on the underlying sample space (and on the measurable space ), one can always find at least one disintegration for every random variable , by using tools such as the Radon-Nikodym theorem; see Theorem 4 of these previous lecture notes of mine. In practice, we will not invoke these general results here (as it is not natural for us to place topological conditions on the sample space), and instead construct disintegrations by hand in specific cases, for instance by using the construction in Example 7.
Remark 8 Strictly speaking, disintegration is not a probabilistic concept; there is no canonical way to extend a disintegration when extending the sample space;. However, due to the (almost) uniqueness and existence results alluded to earlier, this will not be a difficulty in practice. Still, we will try to use conditioning on continuous variables sparingly, in particular containing their use inside the proofs of various lemmas, rather than in their statements, due to their slight incompatibility with the “probabilistic way of thinking”.
Exercise 22 (Fubini-Tonelli theorem) Let be a disintegration of a random variable taking values in a measurable space , and let be a non-negative (resp. absolutely integrable) scalar random variable. Show that for -almost all , is a non-negative (resp. absolutely integrable) random variable, and one has the identity
where is the (almost surely defined) random variable that equals whenever . (Note that one first needs to show that is measurable before one can take the expectation.) More generally, show that
whenever is a non-negative (resp. bounded) measurable function. (One can essentially take (42), together with the fact that is determined by , as a definition of the conditional expectation , but we will not adopt this approach here.)
A typical use of conditioning is to deduce a probabilistic statement from a deterministic one. For instance, suppose one has a random variable , and a parameter in some range , and an event that depends on both and . Suppose we know that for every . Then, we can conclude that whenever is a random variable in independent of , we also have , regardless of what the actual distribution of is. Indeed, if we condition to be a fixed value (using the construction in Example 7, extending the underlying sample space if necessary), we see that for each ; and then one can integrate out the conditioning using (41) to obtain the claim.
The act of conditioning a random variable to be fixed is occasionally also called freezing.
— 5. Convergence —
In a first course in undergraduate real analysis, we learn what it means for a sequence of scalars to converge to a limit ; for every , we have for all sufficiently large . Later on, this notion of convergence is generalised to metric space convergence, and generalised further to topological space convergence; in these generalisations, the sequence can lie in some other space than the space of scalars (though one usually insists that this space is independent of ).
Now suppose that we have a sequence of random variables, all taking values in some space ; we will primarily be interested in the scalar case when is equal to or , but will also need to consider fancier random variables, such as point processes or empirical spectral distributions. In what sense can we say that “converges” to a random variable , also taking values in ?
It turns out that there are several different notions of convergence which are of interest. For us, the four most important (in decreasing order of strength) will be almost sure convergence, convergence in probability, convergence in distribution, and tightness of distribution.
Definition 10 (Modes of convergence) Let be a -compact metric space (with the Borel -algebra), and let be a sequence of random variables taking values in . Let be another random variable taking values in .
- converges almost surely to if, for almost every , converges to , or equivalently
for every .
- converges in probability to if, for every , one has
or equivalently if holds asymptotically almost surely for every .
- converges in distribution to if, for every bounded continuous function , one has
- has a tight sequence of distributions if, for every , there exists a compact subset of such that for all sufficiently large .
Remark 9 One can relax the requirement that be a -compact metric space in the definitions, but then some of the nice equivalences and other properties of these modes of convergence begin to break down. In our applications, though, we will only need to consider the -compact metric space case. Note that all of these notions are probabilistic (i.e. they are preserved under extensions of the sample space).
Exercise 23 (Implications and equivalences) Let be random variables taking values in a -compact metric space .
- (i) Show that if converges almost surely to , then converges in probability to . (Hint: Fatou’s lemma.)
- (ii) Show that if converges in distribution to , then has a tight sequence of distributions.
- (iii) Show that if converges in probability to , then converges in distribution to . (Hint: first show tightness, then use the fact that on compact sets, continuous functions are uniformly continuous.)
- (iv) Show that converges in distribution to if and only if converges to in the vague topology (i.e. for all continuous functions of compact support).
- (v) Conversely, if has a tight sequence of distributions, and is convergent in the vague topology, show that is convergent in distribution to another random variable (possibly after extending the sample space). What happens if the tightness hypothesis is dropped?
- (vi) If is deterministic, show that converges in probability to if and only if converges in distribution to .
- (vii) If has a tight sequence of distributions, show that there is a subsequence of the which converges in distribution. (This is known as Prokhorov’s theorem).
- (viii) If converges in probability to , show that there is a subsequence of the which converges almost surely to .
- (ix) converges in distribution to if and only if for every open subset of , or equivalently if for every closed subset of .
Remark 10 The relationship between almost sure convergence and convergence in probability may be clarified by the following observation. If is a sequence of events, then the indicators converge in probability to zero iff as , but converge almost surely to zero iff as .
Example 9 Let be a random variable drawn uniformly from . For each , let be the event that the decimal expansion of begins with the decimal expansion of , e.g. every real number in lies in . (Let us ignore the annoying ambiguity in the decimal expansion here, as it will almost surely not be an issue.) Then the indicators converge in probability and in distribution to zero, but do not converge almost surely.
If is the digit of , then the converge in distribution (to the uniform distribution on , but do not converge in probability or almost surely. Thus we see that the latter two notions are sensitive not only to the distribution of the random variables, but how they are positioned in the sample space.
The limit of a sequence converging almost surely or in probability is clearly unique up to almost sure equivalence, whereas the limit of a sequence converging in distribution is only unique up to equivalence in distribution. Indeed, convergence in distribution is really a statement about the distributions rather than of the random vaariables themselves. In particular, for convergence in distribution one does not care about how correlated or dependent the are with respect to each other, or with ; indeed, they could even live on different sample spaces and we would still have a well-defined notion of convergence in distribution, even though the other two notions cease to make sense (except when is deterministic, in which case we can recover convergence in probability by Exercise 23(vi)).
Exercise 24 (Borel-Cantelli lemma) Suppose that are random variables such that for every . Show that converges almost surely to .
Exercise 25 (Convergence and moments) Let be a sequence of scalar random variables, and let be another scalar random variable. Let .
- (i) If , show that has a tight sequence of distributions.
- (ii) If and converges in distribution to , show that .
- (iii) If and converges in distribution to , show that .
- (iv) Give a counterexample to show that (iii) fails when , even if we upgrade convergence in distribution to almost sure convergence.
- (v) If the are uniformly bounded and real-valued, and for every , then converges in distribution to . (Hint: use the Weierstrass approximation theorem. Alternatively, use the analytic nature of the moment generating function and analytic continuation.)
- (vi) If the are uniformly bounded and complex-valued, and for every , then converges in distribution to . Give a counterexample to show that the claim fails if one only considers the cases when .
There are other interesting modes of convergence on random variables and on distributions, such as convergence in total variation norm, in the Lévy-Prokhorov metric, or in Wasserstein metric, but we will not need these concepts in this course.
97 comments
Comments feed for this article
2 January, 2010 at 9:33 am
Louigi
Nice notes! Just after Exercise 20, there is a that should probably be a .
[Corrected, thanks - T.]
2 January, 2010 at 4:33 pm
Top Posts — WordPress.com
[...] 254A, Notes 0: A review of probability theory In preparation for my upcoming course on random matrices, I am briefly reviewing some relevant foundational aspects [...] [...]
2 January, 2010 at 4:40 pm
Yihong
In Exercise 23 (v), is the condition that has a tight sequence of distributions superfluous? Because, given that is convergent in the weak-* topology, is tight automatically. Maybe I missed something here…
2 January, 2010 at 6:01 pm
Terence Tao
A weak-* convergent sequence of probability measures is tight if and only if its limit is again a probability measure. If there is loss of mass (consider for instance the uniform distributions on [n,n+1] as ) then one has weak-* convergence to a non-probability measure (zero, in the above example) and lack of tightness.
2 January, 2010 at 5:02 pm
man
Just after eq. 13, should be ?
[Corrected, thanks - T.]
2 January, 2010 at 5:32 pm
links for 2010-01-02 « Blarney Fellow
[...] 254A, Notes 0: A review of probability theory « What’s new (tags: probability math) [...]
3 January, 2010 at 9:40 am
Jason Rute
I noticed the RSS feed for this post didn’t show up in Google Reader (for me anyway). I’m not sure if other people using Google Reader have the same problem. I also wonder if there are other posts I’ve missed because Google Reader skipped them. I know you don’t have any control over this, but I thought I’d let you know.
3 January, 2010 at 1:04 pm
Hassan Mohy-ud-Din
Greetings Professor Terence Tao
Thank You for the notes. Are these notes available in pdf format. Thanking you in anticipation.
With Profound Regards
Good Bye
3 January, 2010 at 3:47 pm
salazar
In the definition of subgaussianness and subexponential tail, it might be more convenient to replace by ?
[Corrected, thanks - T.]
3 January, 2010 at 4:10 pm
timur
Jason,
I think this is not Google Reader, but WordPress. This post did not show up on my RSS reader different than Google’s. I noticed it quite a few times.
3 January, 2010 at 6:34 pm
Steven Heilman
A few typos?–
“is absolutely continuous of”
of -> if
Exercise 18
want with distribution ?
Exercise 19, Eq. 33
[Corrected, thanks - T.]
3 January, 2010 at 10:22 pm
254A, Notes 1: Concentration of measure « What’s new
[...] the Borel-Cantelli lemma (Exercise 1 from Notes 0), we see that we will be done as long as we can choose such [...]
4 January, 2010 at 7:37 pm
Ñie
In the joint random variable example should
$ (X_1, X_2)(\omega) = (X_1(x), X_2(\omega)) $ for every $x \in \Omega $
be
$ (X_1, X_2)(\omega) = (X_1(\omega), X_2(\omega)) $ for every $\omega \in \Omega $ ?
[Corrected, thanks - T.]
5 January, 2010 at 1:58 am
Ñie
there is stiil an error in part. It should be .
[Corrected, thanks - T.]
5 January, 2010 at 2:33 pm
pavel
Dear Professor Tao,
thanks in particular for the illuminating exercises! Some remarks:
Isn’t the identity on [-1 , -0.5] \cup [0.5 , 1] with Lebesgue measure a counterexample to uniqueness in Exercise 7?
I also think that in Definition 10 (convergence in probability) lim sup should be a lim inf.
5 January, 2010 at 3:29 pm
PDEbeginner
Dear Prof. Tao,
It seems the Lemma 5 is not correct.
The (ix) of Ex 23 seems also true for the convergence in distribution sense, since the underlying sample space is not involved.
I don’t understand why (27) needs the condition .
I also don’t know how to obtain an counterexample for the case when in (vi) of Ex 25.
Thanks in advance!
5 January, 2010 at 4:23 pm
Terence Tao
Thanks for the corrections!
The condition a > 1 is needed to ensure that exp(t|X|) is absolutely integrable.
For exercise 25(vi), consider the uniform distribution on the unit circle. More generally, note that if an (absoutely integrable) complex random variable is invariant under multiplication by for any not a k^th root of unity, then its k^{th} moment vanishes.
6 January, 2010 at 8:13 am
Jason
A minor detail: in the definition of modes of convergence, the link for sigma-compact space actually links to Fatou’s lemma on Wikipedia.
Thanks for the great notes! –Jason
6 January, 2010 at 11:59 am
254A, Notes 2: The central limit theorem « What’s new
[...] (ii) is immediate from (6) and the definition of convergence in distribution (see Definition 10 of Notes 0), since the function is bounded [...]
7 January, 2010 at 12:14 am
Manjil P. Saikia
Dear Prof. Tao,
Thanks very much for these notes,they are as always brilliant.
Regards.
7 January, 2010 at 1:18 pm
ANDREWYAKOVLEV
[...] A review of probability theory In preparation for my upcoming course on random matrices, I am briefly reviewing some relevant foundational aspects of probability theory, as well as setting up basic probabilistic notation that we will be using in later posts. This is quite basic material for a graduate course, and somewhat pedantic in nature, but given how heavily we will be relying on probability theory in this course, it seemed appropriate to take some time to go through these issues carefully. [...]
7 January, 2010 at 3:02 pm
Anonymous
in the following sentence
”the event that the roll of a die gives has cardinality one in the sample space , but has cardinality six in the sample space
I think that it should be 12 not 6.
7 January, 2010 at 3:17 pm
Terence Tao
Actually, it is 6. I have modified the text to emphasise that we are talking about the event that the roll of a _given_ die is 4, not that at least one of the dice being rolled is equal to 4; the latter event is not preserved under extensions that increase the number of dice being rolled (and in any case, the latter event would have cardinality 11 rather than 12 in the sample space ).
7 January, 2010 at 3:38 pm
Anonymous
Dear Prof. Tao,
in the proof of lemma 4 we need to show the existence of a probability space on which a random variable has the given distribution.
in the proof, it is not obvious what the is.
7 January, 2010 at 3:44 pm
Terence Tao
is the ambient sample space, which is always assumed to be present whenever we are doing anything with probability theory, and is extended as necessary whenever we introduce more random variables; see the second paragraph of Section 1. (One could explicitly mention “Let be a probability space, with respect to which all random variables mentioned here take as a domain, and all events are measurable in” in the statement of every single lemma and theorem in this course, or in any other text using probabilistic methods, but this would be very tedious, not to mention quite counter to the “probabilistic way of thinking” in which one tries to suppress mention of the sample space as much as possible.)
9 January, 2010 at 12:02 pm
254A, Notes 3: The operator norm of a random matrix « What’s new
[...] we sparsified, it is now safe to apply the Borel-Cantelli lemma (Exercise 1 of Notes 0), and it will suffice to show [...]
9 January, 2010 at 6:11 pm
luca
Great notes! I second the request of making them available in pdf, although they already print rather neatly from wordpress.
By the way, for probability and expectation, you use in some places boldface , , and in some places mathbb , .
9 January, 2010 at 6:39 pm
Terence Tao
Luca: thanks for pointing out the notational inconsistency! (I decided to switch to boldface here, as it seems to be a bit more legible in the blog format.)
I collate the blog files every year into a PDF file; I am working on the 2009 files right now and should have a draft file ready in the near future. Unfortunately, this means that one has to wait for the maximum possible time before this particular post will be converted. (By a strange coincidence, I had almost the exact same question for the Notes 0 article on 1 January 2009.)
9 January, 2010 at 7:23 pm
Pashupati
An article I can somewhat barely understand, at least!
How do you convert those into PDF files? Could anybody help?
And, well, thank you for all those great articles!
10 January, 2010 at 5:10 am
rnd_idt
Simply wait until more or less march 2011 and you will get your PDF. In the meantime, print wordpress, it’s not that bad.
10 January, 2010 at 5:47 pm
CY
You can download and install free Primo PDF converter from the site
http://www.primopdf.com/index.aspx
Then, you just print this page using Primo PDF to get a pdf file.
It works for me, hope it works for u guys too :)
11 January, 2010 at 11:30 am
Anonymous
I have downloaded that programme but could not figure out how it works.
Dear CY, could you explain it?
Thanks
11 January, 2010 at 7:17 pm
CY
Just print this page by selecting Primo PDF as the name of ‘printer’.
10 January, 2010 at 11:47 pm
student
I am a bit surprised by statement in exercise 15 mentioning that for $k>2$, the converse statement fails. If I recall correctly the definition of the k-fold product measure, once we specify the pre-measure on the boxes which is basically done in formula (28), then we get a measure that agrees with the joint distribution measure on the algebra of finite unions of boxes (which generates the full $\sigma-$algebra.) Am I missing something, or does the reason lie somewhere else?
Thanks!
11 January, 2010 at 8:36 am
Terence Tao
There is a subtle difference between (28), (29) and the construct of product measure, having to do with quantifiers.
You may like to think about the k=3 case first. If you know that and , does this imply that are jointly independent?
11 January, 2010 at 6:43 am
Stones Cry Out - If they keep silent… » Things Heard: e101v1
[...] Considering probability. [...]
18 January, 2010 at 6:29 pm
254A, Notes 3b: Brownian motion and Dyson Brownian motion « What’s new
[...] existence of such a collection, after extending the sample space, is guaranteed by Exercise 18 of Notes 0.) If we then [...]
19 January, 2010 at 5:12 pm
Student
I am confused about the existence of random variables. For concreteness fix $\Omega =[0,1]$. Now a real-valued random variable is merely a real valued function on $\Omega$. Let us specify that the random variable $f(x)$ has a Gaussian distribution. Now there is a natural choice for $f(x)$. It isn’t hard to see that I can take $f(x)$ to be a monotonically increasing function such that $f(0)=-\infty$,$f(1)=\infty$. However, does there exists another random variable $g(x)$, that is independent and identically distributed? It seems the answer is ‘no’, since once I know the value of $f(\omega)$ at some choice of $\omega \in \Omega$, I know $\omega$ and hence I know $g(\omega)$.
If I understand this correctly, your proof (exercise 18) that independent identically distributed Gaussian random variables exist may require that you change the underlying probability space $\Omega$. This leads to the following question. Do two independent Gaussian random variables exist on $[0,1]$? If so, is it possible to explicitly describe such a pair?
Any comments that can clear up these general issues would be appreciated.
19 January, 2010 at 5:21 pm
Terence Tao
Well, one can use a space-filling curve to map [0,1] to , and if one does it correctly one can even make the map measure preserving. (e.g. expand [0,1] in base 2 as infinite binary strings, and separate the even and odd binary strings to create a point in ). So one can create two or even countably many variables on the unit interval if one wishes, provided one had enough foresight to do so before starting whatever argument one is working with.
I prefer, though, to just keep extending the probability space whenever necessary, since probability theory is designed to be invariant under this extension operation anyway (see the discussion in Section 1). This way, one does not need to know in advance all the random variables are going to be used in one’s arguments. (To put it more informally: I like to reserve the right to roll more dice whenever I want to.)
21 January, 2010 at 9:00 am
David Speyer
Out of curiosity, is it really a good idea to require in your definition of an extension that $\Omega’ \to \Omega$ be surjective? Why not permit it to miss a set of points of measure 0?
21 January, 2010 at 9:53 am
Terence Tao
My reason for this is that I want to have both “almost surely true” and “surely true” as probabilistic notions. If an extension misses a set of measure zero, then an event which was almost surely, but not surely, true, could become surely true upon taking an extension.
This is not a big deal, of course; for most applications in probability theory there is no significant distinction between almost sure and sure, but I like being able to embed classical logic in probabilistic logic :-)
23 January, 2010 at 6:25 am
Anonymous
Dear Prof. Tao,
If a random variable is uniform (or has any other distribution) on a bounded set, isn’t it surely bounded (and not just a.s. bounded)? (This of course is not too important probabilistically, I am just curious.)
Thank you!
23 January, 2010 at 8:48 am
Terence Tao
The distribution of a random variable X only gives the probability that X lies in various sets, but cannot force X to lie in a set surely. For instance, if we took a uniform distribution on [0,1] and modified it on an event of probability zero to equal 2 (say) on that event, then the new random variable Y would still be considered to have the uniform distribution on [0,1], since is still given by Lebesgue measure on [0,1].
27 January, 2010 at 9:38 am
Dmitry Karpeev
Prof. Tao,
If I understand it correctly, Exercise 18 is equivalent supplying a proof of the existence of a probability measure on a suitable infinite product space, which reduces to a given probability measure \mu_\beta when restricted to the factor with index \beta. Is this so, or can the result of Exercise 18 be established by more elementary means?
Thank you!
Dmitry.
27 January, 2010 at 8:34 pm
Terence Tao
Yes, the ability to construct arbitrarily many independent random variables with specified distribution in probability theory is essentially equivalent to the ability to construct product measures in measure theory.
In the case when the desired random variables are real-valued, and there are at most countably many of them, one can use more ad hoc but concrete methods (e.g. using a uniform distribution on [0,1] and converting into binary to create an infinite stream of bits, and subdividing them to create as many sources of randomness as necessary) to create the random variables.
One can also proceed via the Caratheodory extension theorem or Kolmogorov extension theorem in many cases (but this is how product measure is usually constructed anyway; another approach is via the Riesz representation theorem).
30 January, 2010 at 4:29 pm
Mio
typo: in the real normal distribution example there’s a 2 missing in the denominator of the exponent, next to
[Corrected, thanks -T.]
30 January, 2010 at 7:08 pm
Ultralimit analysis, and quantitative algebraic geometry « What’s new
[...] of the reals as much as possible. Similarly for other rows of the above table. See for instance lecture notes of mine for further discussion of the distinction between measure theory and probability [...]
2 February, 2010 at 1:34 pm
254A, Notes 4: The semi-circular law « What’s new
[...] we consider the behaviour of the ESD of a sequence of Hermitian matrix ensembles as . Recall from Notes 0 that for any sequence of random variables in a -compact metrisable space, one can define notions of [...]
4 February, 2010 at 12:32 am
Anonymous
Section 1, paragraph 1, line -1: “will will” -> “will”
[Corrected, thanks - T.]
10 February, 2010 at 10:56 pm
245A, Notes 5: Free probability « What’s new
[...] as possible, and with the random variables and expectations being viewed as derived concepts. See Notes 0 for further discussion of this [...]
16 February, 2010 at 9:15 pm
Mio
typo: in Definition 1. first occurrence of should be
[Corrected, thanks - T.]
20 February, 2010 at 7:59 am
Afonso
Dear Prof. Tao
I don’t think I fully understand the meaning of with probability greater than p.
Does it mean that there exists a constant C such that for every n
with probability larger than p.
Or does it mean that if we consider the whole sequence as a random variable (a stochastic process) the output sequence is O(1) with probability greater than p?
It is not clear to me whether the two are equivalent.
Thank You,
Afonso
20 February, 2010 at 8:36 am
Terence Tao
I am using “For all , with probability at least ” to denote the statement , with the (much) stronger statement being denoted instead by “With probability at least , one has for all “.
In most cases, the universal quantification over appears very early in a statement or argument (or is implicit in phrases such as “let n be an integer parameter”), and so one usually is referring to the first type of bound. The second type usually only comes up in this course in discussions of almost sure convergence, and is usually derived from estimates of the first type using the union bound or the Borel-Cantelli lemma (perhaps after first sparsifying n to a lacunary sequence).
25 February, 2010 at 12:50 am
Sam Watson
In Exercise 4 (the first part), I think that a mean-zero condition on the random variables is required. Consider the random variable which is almost surely equal to 1. It is subgaussian according to the definition given, since the indicator of [0,1] is dominated by a scalar multiple of a gaussian. On the other hand, there is no C for which the inequality in Exercise 4 will be true, since the identity function on the reals is not dominated by any parabola with vertex at the origin.
25 February, 2010 at 9:21 am
Terence Tao
Ah, right, I was missing an additional factor of C in front of the factor. I think I’ve fixed the issue now.
8 March, 2010 at 6:18 am
Anonymous
In Definition 10, it should be liminf instead of limsup.
14 March, 2010 at 7:21 am
Anonymous
If X_n converges in probability to X , show that there is a subsequence of the which converges almost surely to X.
ANy hint or reference to this result, I could not figure it out…
14 March, 2010 at 10:18 am
Terence Tao
Find a subsequence so that for every , then apply the Borel-Cantelli lemma.
26 September, 2010 at 12:22 pm
245A, Notes 3: Integration on abstract measure spaces, and the convergence theorems « What’s new
[...] objects of study of that theory, namely the behaviour of random events and random variables. (See this post for further discussion of this point.) This course will not be focused on applications to [...]
24 November, 2010 at 6:01 am
Guillaume
Dear Prof. Tao,
I’m a bit puzzled by Exercise 25 (v) [if a sequence of uniformly sub-exponential r.v. converges in moments, then it converges in distribution], since I have the impression that sub-exponential is not a strong enough property to be determined by the moments ; for example there are counterexamples to the moments problem with density $\latex \exp(-|x|^{1/2+o(1)})$. Did I miss something ?
[Ah, I had omitted the requirement that the exponent is at least 1. Thanks - T.]
25 November, 2010 at 1:55 am
Guillaume
OK, I see !
How do you prove this using truncation and the Weierstrass approximation theorem? My impression is that you need some quantitative information, such as the speed of convergence in the Weierstrass approximation.
25 November, 2010 at 10:53 am
Terence Tao
Hmm, good point; the argument I had in mind would indeed require a quantitative version of the Weierstrass approximation theorem. I guess for the purposes of this exercise it is probably best to just handle the uniformly bounded case.
22 November, 2011 at 11:10 am
Noam Zeilberger
typo in Proof of Lemma 4? “the product measure on $R$” should be “the product measure on ”
[Corrected, thanks - T]
2 January, 2012 at 12:16 pm
Sean Eberhard
Dear Prof Tao,
This introductory to probability theory does a great job of clarifying the distinction of probability theory from the theory of finite measures. Do any other authors that you know of make as explicit the notion of invariance under extensions?
By the way, like a previous commenter, I too thought that excluding surjectivity from the definition of extension would be a neat way of making “almost surely” a more intrinsic part of the theory. Remark 1 is simplified in this situation, because all random variables would be only almost surely defined. Conditioning is also somewhat unified with extending, because then is an extension of with respect to the probability measure .
26 February, 2012 at 3:05 pm
Rex
Small typo: Exercise 9 reads
“Show that the {(X_\alpha)_{\alpha \in A}} are jointly independent if and only for every…”
Of course, this should say “if and only if”
[Corrected, thanks - T.]
26 February, 2012 at 3:23 pm
Rex
Also, Exercise 10 has the typo “subsapce”.
[Corrected, thanks - T.]
2 April, 2012 at 4:55 pm
A cheap version of nonstandard analysis « What’s new
[...] To set up cheap nonstandard analysis, we will need an asymptotic parameter , which we will take to initially lie in the natural numbers (though it is certainly possible to set up cheap nonstandard analysis on other non-compact spaces than if one wishes). However, we reserve the right in the future to restrict the parameter space from to a smaller infinite subset (which corresponds to the familiar operation of passing from a sequence to a subsequence, except that we do not bother to relabel the subsequence to be indexed by again). The dynamic nature of the parameter space makes it a little tricky to properly formalise cheap nonstandard analysis in the usual static framework of mathematical logic, but it turns out not to make much difference in practice, because in cheap nonstandard analysis one only works with statements which remain valid under the operation of restricting the underlying domain of the asymptotic parameter. (This is analogous to how in probability theory one only works with statements which remain valid under the operation of extending the underlying probability space, as discussed in this blog post.) [...]
4 April, 2012 at 5:45 am
Musfir
Reblogged this on Geekmusfir.
11 April, 2012 at 6:29 pm
Rex
Given two (or more) independent random variables , can we always lift the sample space to some which can factored as a product of measure spaces (with product measure) such that is defined on and is defined on ?
The definition of independence ensures that we can do this for the corresponding distributions, but can we do it directly on the sample space?
As an informal guideline, if we pretend that the sample space breaks into a product measure space whenever we see independent random variables, will we ever get into trouble while working in probability?
11 April, 2012 at 8:36 pm
Terence Tao
Yes, this should be true (at least under some mild assumptions on the sigma algebras involved (e.g. if all the underlying spaces are standard Borel), so that tools such as disintegration become available. But one should caution that the product structure may not be canonical or unique. For instance, Example 3 above can be given three different product structures, each of which makes two of the three variables manifestly independent, but the three structures are not compatible with each other. So it is perhaps best not to rely on this viewpoint too strongly.
11 April, 2012 at 8:44 pm
Rex
Do concerns about lack of canonicity or uniqueness with regards to the sample space ever disturb us as probabilists?
It seems that we de-emphasize the role of the sample space so much that it is not clear to me how its universal properties could play any role in probability.
My general aim is to grasp better this “probabilistic” way of thinking, in particular to understand in what ways it allows us to replace or modify the sample space without affecting our tools.
11 April, 2012 at 7:01 pm
Rex
When defining real-valued random variables, does one typically equip the real line with the Borel measure, or its Lebesgue completion? Does this distinction matter much in practice?
For instance, does one have to do a significant amount of extra work to check that certain random variables are Lebesgue-measurable as opposed to merely Borel-measurable?
In the definition you refer only to the Borel measure, but later on you mention some issues about pullbacks of (Lebesgue) null sets when discussing absolute continuity of random variables.
11 April, 2012 at 8:28 pm
Terence Tao
In general, the Borel sigma algebra is slightly more convenient to use than the Lebesgue sigma algebra for the _range_ of a measurable function, but the Lebesgue can be more a convenient algebra to use for the _domain_ of a measurable function. But the main advantage of Lebesgue measure, namely completeness, is more useful in measure theory than in probability theory; for most probabilistic applications one does not actually need completeness.
p.s. I don’t know what issue about pullbacks of null sets you are referring to in your comment.
11 April, 2012 at 8:32 pm
Rex
I did not really mean to say there was any “issue”, but rather just that you switched from Borel measure to Lebesgue measure in the following passage:
“Now we turn to non-discrete random variables {X} taking values in some range {R}. We say that a random variable is continuous if {{\bf P}(X=x)=0} f
or all {x \in R} (here we assume that all points are measurable). If {R} is already equipped with some reference measure {dm} (e.g. Lebesgue measure in the case of scalar, vector, or matrix-valued random variables), we say that the random variable is absolutely continuous if {{\bf P}(X \in S)=0} for all null sets {S} in {R}. ”
and it was not clear to me whether there was any significance in this switch.
11 April, 2012 at 8:43 pm
Terence Tao
Ah. I tend to use Lebesgue measure to denote both the standard measure on the Lebesgue sigma algebra, as well as its restriction to the Borel sigma algebra (which is indeed the slightly more natural sigma algebra to use in this context). (The terminology “the Borel measure on ” to denote this restriction is also in use, but somewhat less common, perhaps because it can be confused with the more general concept of a Borel measure.)
17 June, 2012 at 9:41 am
frankpmurphyh
Reblogged this on algebrafm.
5 September, 2012 at 6:12 pm
Gelasio Salazar
Dear Terry,
One question about the distinction between “with high probability” and
“asymptotically almost surely”. We just got a referee report in which they
ask us to change “w.h.p.” to “a.a.s.” — since in a particular lemma, all
we can prove is that a certain event holds with probability 1 -o(1). I
would have normally used w.h.p. and a.a.s. interchangeably, but after the
referee’s remark (s/he gave your Notes as reference) I realized we need to
be more careful. In the revised version we’ll use “a.a.s.”, and I was
wondering if you were aware of other sources in which this distinction
between “overwhelming probability”, “with high probability” and
“asymptotically almost surely” is used.
Last but not least, thanks for your comprehensive notes in Probability
Theory.
5 September, 2012 at 8:21 pm
Terence Tao
“asymptotically almost surely”, when it is used in literature, invariably means 1-o(1), but for “with high probability” there is less consensus; I have seen it used for both 1-o(1) and for 1-O(n^{-c}) (though not in the same paper, of course). But given that a.a.s. is a perfectly useful and accepted notation for 1-o(1), it seems logical to me to exclusively use w.h.p for 1-O(n^{-c}) instead.
19 September, 2012 at 10:59 am
Jack
Could you give an example of your saying that “If one was particularly well organised, one could in principle work out in advance all of the random variables one would ever want or need, and then specify the sample space accordingly, before doing any actual probability theory.”?
19 September, 2012 at 11:28 am
Jack
Can one say to some degree that a random variable on can be regarded as an extension ?
27 September, 2012 at 5:24 pm
Jack
I’m confused about the concept “pushforward”. Let be a probability space and random variable on this space with range . Then is a probability measure on . However, according to your notes of 245A, can be any measurable space, for example . But I also learned that one cannot sign a measure to that render it a measure space. What do I do wrong here?
28 September, 2012 at 3:08 am
Terence Tao
One can place several measures on , e.g. a Dirac measure. (But one cannot have a non-trivial translation-invariant measure on this space, due to Banach-Tarski type paradoxes.)
28 September, 2012 at 6:02 am
Jack
Ah, I see the point. As you said in this note, the underlying sample space of a random variable is often not specified. And the range of the random variable , according to Remark 2, can be somehow not mentioned either as I understand. I’m puzzled about this: to what extend should one specify a random variable? What’s left for a function when one does not specify its domain and range?
I saw lots of times when one says something like “consider a -value random variable”. They don’t even specify which algebra is used for . What’s more, I’ve never read that one specify a measure for the range measurable space . Is it because we have an immediate one, , or it doesn’t matter at all?
28 September, 2012 at 12:27 pm
Terence Tao
We usually don’t specify the sample space of a random variable for much the same reason we don’t specify which base (e.g. base 10, binary, etc.) we use to represent numbers, or which coordinate system we use to represent a manifold. We could specify these representations, if desired, but we wish to focus on those aspects of the mathematical objects being studied that are independent of the choice of representation, and so it is usually counterproductive to devote too much attention to these representations. (This is discussed near the beginning of this blog post.)
Also, one should make a distinction between a measurable space and a measure space . A measurable space can be turned into a measure space by specifying a measure, but there are multiple measures one could use for this purpose, and in many cases it is better to not specify a measure at all.
15 October, 2012 at 9:01 am
SAADA
Professeur Tao, auriez-vous une version française de cette note ?
Merci par avance.
15 October, 2012 at 10:08 am
Terence Tao
Non, mais outils de traduction automatique, tels que Google Translate, peuvent faire un travail raisonnable: http://translate.google.com/translate?sl=en&tl=fr&js=n&prev=_t&hl=en&ie=UTF-8&layout=2&eotf=1&u=http%3A%2F%2Fterrytao.wordpress.com%2F2010%2F01%2F01%2F254a-notes-0-a-review-of-probability-theory&act=url
15 October, 2012 at 10:35 am
Daniel
merci beaucoup professeur.
http://www.daniel-saada.eu
15 October, 2012 at 10:48 pm
Hoeffding bound | blayz
[...] first two sections of the book Topics in Random Matrix Theory by Terry Tao, with his draft and relevant notes available online, and the Hoeffding’s original paper along with the paper by [...]
4 January, 2013 at 9:45 am
http terrytao wordpress com 2010 01 01 254a… « Kathys LinkBook
[...] http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ [...]
19 January, 2013 at 12:14 pm
tim
Hi Prof Tao,
Thanks for sharing your note!
I don’t quite understand why the extension is defined to be surjective in Tao’s blog. Is the concept “extension” trying to become morphisms in some category of probability spaces?
I found two links http://theoreticalatlas.wordpress.com/2010/11/11/categorifying-measure-theory/ and http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th. Although I can only understand some small part of them, I hope they can be interesting to you!
Thanks!
23 March, 2013 at 5:05 am
Marek
Dear Prof. Tao,
Excellent post, thank you for sharing! I have a related question, and I would be very grateful if you or someone else could provide me an answer.
As is well-known, the total variation distance between (the laws of) two random variables X and Y defined on R is given by sup|E[g(X)]−E[g(Y)]|, where the supremum is taken over all g:R→[0,1] that are measurable. Here is my question: do we obtain the same definition if one considers the supremum over all g:R→[0,1] that are continuous? Or over all g:R→[0,1] that are differentiable?
Thanks a lot in advance!
23 March, 2013 at 7:11 am
Terence Tao
Yes, basically because test functions are dense in for any Borel probability measure on the real line (note that all Borel probability measures on R are Radon measures), thanks to basic tools such as the Stone-Weierstrass theorem, Lusin’s theorem, the Tietze extension theorem, and Urysohn’s lemma (as discussed in this previous blog post). Applying this general fact to , the Borel probability measure formed by the average of the two laws, and using the Radon-Nikodym theorem to write as bounded multiples of , we obtain the desired equivalence.
23 March, 2013 at 7:56 am
Marek
Thank you very much for the prompt reply! It is very clear in my mind now. Best, Marek
11 July, 2013 at 11:53 am
Probability and Statistics Books Online | Download free ebook
[…] A review of probability theory by Terence Tao, 2010 […]
16 September, 2013 at 7:12 am
Free Mathematics eBooks Online : Probability and Statistics | Top Free Books
[…] A review of probability theory by Terence Tao, 2010 […]
16 November, 2013 at 10:12 pm
Qualitative probability theory, types, and the group chunk and group configuration theorems | What's new
[…] classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space – a space (the sample space) equipped with […]
16 November, 2013 at 10:12 pm
Qualitative probability theory, types, and the group chunk and group configuration theorems | What's new
[…] classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space – a space (the sample space) equipped with […]