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Let be the Liouville function, thus is defined to equal when is the product of an even number of primes, and when is the product of an odd number of primes. The Chowla conjecture asserts that has the statistics of a random sign pattern, in the sense that

for all and all distinct natural numbers , where we use the averaging notation

For , this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any .

In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version

of the conjecture, where we use the logarithmic averaging notation

Using the summation by parts (or telescoping series) identity

it is not difficult to show that the Chowla conjecture (1) for a given implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for , we have already mentioned that the Chowla conjecture

is equivalent to the prime number theorem; but the logarithmically averaged analogue

is significantly easier to show (a proof with the Liouville function replaced by the closely related Möbius function is given in this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for , and in this recent paper with Joni Teravainen, we proved the conjecture for all odd (with a different proof also given here).

In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:

Theorem 1Assume that the logarithmically averaged Chowla conjecture (2) is true for all . Then there exists a sequence going to infinity such that the Chowla conjecture (1) is true for all along that sequence, that is to sayfor all and all distinct .

This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.

On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:

Theorem 2Let be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for . Then there exists a set of natural numbers of logarithmic density (that is, ) such thatfor any distinct .

It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture ( and odd ) are currently insufficient to use Theorem 2 for any purpose other than to reprove what is already known to be true from the prime number theorem. (Indeed, the even cases of Chowla, in either logarithmically averaged or non-logarithmically averaged forms, seem to be far more powerful than the odd cases; see Remark 1.7 of this paper of myself and Teravainen for a related observation in this direction.)

We now sketch the proof of Theorem 2. For any distinct , we take a large number and consider the limiting the second moment

We can expand this as

If all the are distinct, the hypothesis (2) tells us that the inner averages goes to zero as . The remaining averages are , and there are of these averages. We conclude that

By Markov’s inequality (and (3)), we conclude that for any fixed , there exists a set of upper logarithmic density at least , thus

such that

By deleting at most finitely many elements, we may assume that consists only of elements of size at least (say).

For any , if we let be the union of for , then has logarithmic density . By a diagonalisation argument (using the fact that the set of tuples is countable), we can then find a set of natural numbers of logarithmic density , such that for every , every sufficiently large element of lies in . Thus for every sufficiently large in , one has

for some with . By Cauchy-Schwarz, this implies that

interchanging the sums and using and , this implies that

We conclude on taking to infinity that

as required.

Suppose we have an matrix that is expressed in block-matrix form as

where is an matrix, is an matrix, is an matrix, and is a matrix for some . If is invertible, we can use the technique of Schur complementation to express the inverse of (if it exists) in terms of the inverse of , and the other components of course. Indeed, to solve the equation

where are column vectors and are column vectors, we can expand this out as a system

Using the invertibility of , we can write the first equation as

and substituting this into the second equation yields

and thus (assuming that is invertible)

and then inserting this back into (1) gives

Comparing this with

we have managed to express the inverse of as

One can consider the inverse problem: given the inverse of , does one have a nice formula for the inverse of the minor ? Trying to recover this directly from (2) looks somewhat messy. However, one can proceed as follows. Let denote the matrix

(with the identity matrix), and let be its transpose:

Then for any scalar (which we identify with times the identity matrix), one has

and hence by (2)

noting that the inverses here will exist for large enough. Taking limits as , we conclude that

On the other hand, by the Woodbury matrix identity (discussed in this previous blog post), we have

and hence on taking limits and comparing with the preceding identity, one has

This achieves the aim of expressing the inverse of the minor in terms of the inverse of the full matrix. Taking traces and rearranging, we conclude in particular that

In the case, this can be simplified to

where is the basis column vector.

We can apply this identity to understand how the spectrum of an random matrix relates to that of its top left minor . Subtracting any complex multiple of the identity from (and hence from ), we can relate the Stieltjes transform of with the Stieltjes transform of :

At this point we begin to proceed informally. Assume for sake of argument that the random matrix is Hermitian, with distribution that is invariant under conjugation by the unitary group ; for instance, could be drawn from the Gaussian Unitary Ensemble (GUE), or alternatively could be of the form for some real diagonal matrix and a unitary matrix drawn randomly from using Haar measure. To fix normalisations we will assume that the eigenvalues of are typically of size . Then is also Hermitian and -invariant. Furthermore, the law of will be the same as the law of , where is now drawn uniformly from the unit sphere (independently of ). Diagonalising into eigenvalues and eigenvectors , we have

One can think of as a random (complex) Gaussian vector, divided by the magnitude of that vector (which, by the Chernoff inequality, will concentrate to ). Thus the coefficients with respect to the orthonormal basis can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for distance away from the real axis at least) that one has the concentration of measure

and thus

(that is to say, the diagonal entries of are roughly constant). Similarly we have

Inserting this into (5) and discarding terms of size , we thus conclude the approximate relationship

This can be viewed as a difference equation for the Stieltjes transform of top left minors of . Iterating this equation, and formally replacing the difference equation by a differential equation in the large limit, we see that when is large and for some , one expects the top left minor of to have Stieltjes transform

where solves the Burgers-type equation

Example 1If is a constant multiple of the identity, then . One checks that is a steady state solution to (7), which is unsurprising given that all minors of are also times the identity.

Example 2If is GUE normalised so that each entry has variance , then by the semi-circular law (see previous notes) one has (using an appropriate branch of the square root). One can then verify the self-similar solutionto (7), which is consistent with the fact that a top minor of also has the law of GUE, with each entry having variance when .

One can justify the approximation (6) given a sufficiently good well-posedness theory for the equation (7). We will not do so here, but will note that (as with the classical inviscid Burgers equation) the equation can be solved exactly (formally, at least) by the method of characteristics. For any initial position , we consider the characteristic flow formed by solving the ODE

with initial data , ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation (7) implies that

and thus . Inserting this back into (8) we see that

and thus (7) may be solved implicitly via the equation

Remark 3In practice, the equation (9) may stop working when crosses the real axis, as (7) does not necessarily hold in this region. It is a cute exercise (ultimately coming from the Cauchy-Schwarz inequality) to show that this crossing always happens, for instance if has positive imaginary part then necessarily has negative or zero imaginary part.

Example 4Suppose we have as in Example 1. Then (9) becomesfor any , which after making the change of variables becomes

as in Example 1.

Example 5Suppose we haveas in Example 2. Then (9) becomes

If we write

one can calculate that

and hence

One can recover the spectral measure from the Stieltjes transform as the weak limit of as ; we write this informally as

In this informal notation, we have for instance that

which can be interpreted as the fact that the Cauchy distributions converge weakly to the Dirac mass at as . Similarly, the spectral measure associated to (10) is the semicircular measure .

If we let be the spectral measure associated to , then the curve from to the space of measures is the high-dimensional limit of a Gelfand-Tsetlin pattern (discussed in this previous post), if the pattern is randomly generated amongst all matrices with spectrum asymptotic to as . For instance, if , then the curve is , corresponding to a pattern that is entirely filled with ‘s. If instead is a semicircular distribution, then the pattern is

thus at height from the top, the pattern is semicircular on the interval . The interlacing property of Gelfand-Tsetlin patterns translates to the claim that (resp. ) is non-decreasing (resp. non-increasing) in for any fixed . In principle one should be able to establish these monotonicity claims directly from the PDE (7) or from the implicit solution (9), but it was not clear to me how to do so.

An interesting example of such a limiting Gelfand-Tsetlin pattern occurs when , which corresponds to being , where is an orthogonal projection to a random -dimensional subspace of . Here we have

and so (9) in this case becomes

A tedious calculation then gives the solution

For , there are simple poles at , and the associated measure is

This reflects the interlacing property, which forces of the eigenvalues of the minor to be equal to (resp. ). For , the poles disappear and one just has

For , one has an inverse semicircle distribution

There is presumably a direct geometric explanation of this fact (basically describing the singular values of the product of two random orthogonal projections to half-dimensional subspaces of ), but I do not know of one off-hand.

The evolution of can also be understood using the *-transform* and *-transform* from free probability. Formally, letlet be the inverse of , thus

for all , and then define the -transform

The equation (9) may be rewritten as

and hence

See these previous notes for a discussion of free probability topics such as the -transform.

Example 6If then the transform is .

Example 7If is given by (10), then the transform is

Example 8If is given by (11), then the transform is

This simple relationship (12) is essentially due to Nica and Speicher (thanks to Dima Shylakhtenko for this reference). It has the remarkable consequence that when is the reciprocal of a natural number , then is the free arithmetic mean of copies of , that is to say is the free convolution of copies of , pushed forward by the map . In terms of random matrices, this is asserting that the top minor of a random matrix has spectral measure approximately equal to that of an arithmetic mean of independent copies of , so that the process of taking top left minors is in some sense a continuous analogue of the process of taking freely independent arithmetic means. There ought to be a geometric proof of this assertion, but I do not know of one. In the limit (or ), the -transform becomes linear and the spectral measure becomes semicircular, which is of course consistent with the free central limit theorem.

In a similar vein, if one defines the function

and inverts it to obtain a function with

for all , then the *-transform* is defined by

Writing

for any , , we have

and so (9) becomes

which simplifies to

replacing by we obtain

and thus

and hence

One can compute to be the -transform of the measure ; from the link between -transforms and free products (see e.g. these notes of Guionnet), we conclude that is the free product of and . This is consistent with the random matrix theory interpretation, since is also the spectral measure of , where is the orthogonal projection to the span of the first basis elements, so in particular has spectral measure . If is unitarily invariant then (by a fundamental result of Voiculescu) it is asymptotically freely independent of , so the spectral measure of is asymptotically the free product of that of and of .

In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices. I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in random matrix theory; this is a slightly different set of topics than I had initially advertised (which was instead about the Lindeberg exchange method and the local relaxation flow method), but after consulting with the other mini-course lecturers I felt that this would be a more complementary set of topics. I have uploaded an draft of my lecture notes (some portion of which is derived from my monograph on the subject); as always, comments and corrections are welcome.

<I>[Update, June 23: notes revised and reformatted to PCMI format. -T.]</I>

Let be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic

where denotes the estimate as . Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic number theory estimates discussed here) through the probabilistic lens. Namely, if is a random number selected uniformly between and , then the above estimate can be written as

that is to say the random variable has mean approximately . (But, somewhat paradoxically, this is not the median or mode behaviour of this random variable, which instead concentrates near , basically thanks to the Hardy-Ramanujan theorem.)

Now we turn to the pair correlations for a fixed positive integer . There is a classical computation of Ingham that shows that

The error term in (2) has been refined by many subsequent authors, as has the uniformity of the estimates in the aspect, as these topics are related to other questions in analytic number theory, such as fourth moment estimates for the Riemann zeta function; but we will not consider these more subtle features of the estimate here. However, we will look at the next term in the asymptotic expansion for (2) below the fold.

Using our probabilistic lens, the estimate (2) can be written as

From (1) (and the asymptotic negligibility of the shift by ) we see that the random variables and both have a mean of , so the additional factor of represents some arithmetic coupling between the two random variables.

Ingham’s formula can be established in a number of ways. Firstly, one can expand out and use the hyperbola method (splitting into the cases and and removing the overlap). If one does so, one soon arrives at the task of having to estimate sums of the form

for various . For much less than this can be achieved using a further application of the hyperbola method, but for comparable to things get a bit more complicated, necessitating the use of non-trivial estimates on Kloosterman sums in order to obtain satisfactory control on error terms. A more modern approach proceeds using automorphic form methods, as discussed in this previous post. A third approach, which unfortunately is only heuristic at the current level of technology, is to apply the Hardy-Littlewood circle method (discussed in this previous post) to express (2) in terms of exponential sums for various frequencies . The contribution of “major arc” can be computed after a moderately lengthy calculation which yields the right-hand side of (2) (as well as the correct lower order terms that are currently being suppressed), but there does not appear to be an easy way to show directly that the “minor arc” contributions are of lower order, although the methods discussed previously do indirectly show that this is ultimately the case.

Each of the methods outlined above requires a fair amount of calculation, and it is not obvious while performing them that the factor will emerge at the end. One can at least explain the as a normalisation constant needed to balance the factor (at a heuristic level, at least). To see this through our probabilistic lens, introduce an independent copy of , then

using symmetry to order (discarding the diagonal case ) and making the change of variables , we see that (4) is heuristically consistent with (3) as long as the asymptotic mean of in is equal to . (This argument is not rigorous because there was an implicit interchange of limits present, but still gives a good heuristic “sanity check” of Ingham’s formula.) Indeed, if denotes the asymptotic mean in , then we have (heuristically at least)

and we obtain the desired consistency after multiplying by .

This still however does not explain the presence of the factor. Intuitively it is reasonable that if has many prime factors, and has a lot of factors, then will have slightly more factors than average, because any common factor to and will automatically be acquired by . But how to quantify this effect?

One heuristic way to proceed is through analysis of local factors. Observe from the fundamental theorem of arithmetic that we can factor

where the product is over all primes , and is the local version of at (which in this case, is just one plus the –valuation of : ). Note that all but finitely many of the terms in this product will equal , so the infinite product is well-defined. In a similar fashion, we can factor

where

(or in terms of valuations, ). Heuristically, the Chinese remainder theorem suggests that the various factors behave like independent random variables, and so the correlation between and should approximately decouple into the product of correlations between the local factors and . And indeed we do have the following local version of Ingham’s asymptotics:

Proposition 1 (Local Ingham asymptotics)For fixed and integer , we haveand

From the Euler formula

we see that

and so one can “explain” the arithmetic factor in Ingham’s asymptotic as the product of the arithmetic factors in the (much easier) local Ingham asymptotics. Unfortunately we have the usual “local-global” problem in that we do not know how to rigorously derive the global asymptotic from the local ones; this problem is essentially the same issue as the problem of controlling the minor arc contributions in the circle method, but phrased in “physical space” language rather than “frequency space”.

Remark 2The relation between the local means and the global mean can also be seen heuristically through the applicationof Mertens’ theorem, where is Pólya’s magic exponent, which serves as a useful heuristic limiting threshold in situations where the product of local factors is divergent.

Let us now prove this proposition. One could brute-force the computations by observing that for any fixed , the valuation is equal to with probability , and with a little more effort one can also compute the joint distribution of and , at which point the proposition reduces to the calculation of various variants of the geometric series. I however find it cleaner to proceed in a more recursive fashion (similar to how one can prove the geometric series formula by induction); this will also make visible the vague intuition mentioned previously about how common factors of and force to have a factor also.

It is first convenient to get rid of error terms by observing that in the limit , the random variable converges vaguely to a uniform random variable on the profinite integers , or more precisely that the pair converges vaguely to . Because of this (and because of the easily verified uniform integrability properties of and their powers), it suffices to establish the exact formulae

in the profinite setting (this setting will make it easier to set up the recursion).

We begin with (5). Observe that is coprime to with probability , in which case is equal to . Conditioning to the complementary probability event that is divisible by , we can factor where is also uniformly distributed over the profinite integers, in which event we have . We arrive at the identity

As and have the same distribution, the quantities and are equal, and (5) follows by a brief amount of high-school algebra.

We use a similar method to treat (6). First treat the case when is coprime to . Then we see that with probability , and are simultaneously coprime to , in which case . Furthermore, with probability , is divisible by and is not; in which case we can write as before, with and . Finally, in the remaining event with probability , is divisible by and is not; we can then write , so that and . Putting all this together, we obtain

and the claim (6) in this case follows from (5) and a brief computation (noting that in this case).

Now suppose that is divisible by , thus for some integer . Then with probability , and are simultaneously coprime to , in which case . In the remaining event, we can write , and then and . Putting all this together we have

which by (5) (and replacing by ) leads to the recursive relation

and (6) then follows by induction on the number of powers of .

The estimate (2) of Ingham was refined by Estermann, who obtained the more accurate expansion

for certain complicated but explicit coefficients . For instance, is given by the formula

where is the Euler-Mascheroni constant,

The formula for is similar but even more complicated. The error term was improved by Heath-Brown to ; it is conjectured (for instance by Conrey and Gonek) that one in fact has square root cancellation here, but this is well out of reach of current methods.

These lower order terms are traditionally computed either from a Dirichlet series approach (using Perron’s formula) or a circle method approach. It turns out that a refinement of the above heuristics can also predict these lower order terms, thus keeping the calculation purely in physical space as opposed to the “multiplicative frequency space” of the Dirichlet series approach, or the “additive frequency space” of the circle method, although the computations are arguably as messy as the latter computations for the purposes of working out the lower order terms. We illustrate this just for the term below the fold.

Note: the following is a record of some whimsical mathematical thoughts and computations I had after doing some grading. It is likely that the sort of problems discussed here are in fact well studied in the appropriate literature; I would appreciate knowing of any links to such.

Suppose one assigns true-false questions on an examination, with the answers randomised so that each question is equally likely to have “true” as the correct answer as “false”, with no correlation between different questions. Suppose that the students taking the examination must answer each question with exactly one of “true” or “false” (they are not allowed to skip any question). Then it is easy to see how to grade the exam: one can simply count how many questions each student answered correctly (i.e. each correct answer scores one point, and each incorrect answer scores zero points), and give that number as the final grade of the examination. More generally, one could assign some score of points to each correct answer and some score (possibly negative) of points to each incorrect answer, giving a total grade of points. As long as , this grade is simply an affine rescaling of the simple grading scheme and would serve just as well for the purpose of evaluating the students, as well as encouraging each student to answer the questions as correctly as possible.

In practice, though, a student will probably not know the answer to each individual question with absolute certainty. One can adopt a probabilistic model, where for a given student and a given question , the student may think that the answer to question is true with probability and false with probability , where is some quantity that can be viewed as a measure of confidence has in the answer (with being confident that the answer is true if is close to , and confident that the answer is false if is close to ); for simplicity let us assume that in ‘s probabilistic model, the answers to each question are independent random variables. Given this model, and assuming that the student wishes to maximise his or her expected grade on the exam, it is an easy matter to see that the optimal strategy for to take is to answer question true if and false if . (If , the student can answer arbitrarily.)

[Important note: here we are *not* using the term “confidence” in the technical sense used in statistics, but rather as an informal term for “subjective probability”.]

This is fine as far as it goes, but for the purposes of evaluating how well the student actually knows the material, it provides only a limited amount of information, in particular we do not get to directly see the student’s subjective probabilities for each question. If for instance answered out of questions correctly, was it because he or she actually knew the right answer for seven of the questions, or was it because he or she was making educated guesses for the ten questions that turned out to be slightly better than random chance? There seems to be no way to discern this if the only input the student is allowed to provide for each question is the single binary choice of true/false.

But what if the student were able to give probabilistic answers to any given question? That is to say, instead of being forced to answer just “true” or “false” for a given question , the student was allowed to give answers such as “ confident that the answer is true” (and hence confidence the answer is false). Such answers would give more insight as to how well the student actually knew the material; in particular, we would theoretically be able to actually see the student’s subjective probabilities .

But now it becomes less clear what the right grading scheme to pick is. Suppose for instance we wish to extend the simple grading scheme in which an correct answer given in confidence is awarded one point. How many points should one award a correct answer given in confidence? How about an incorrect answer given in confidence (or equivalently, a correct answer given in confidence)?

Mathematically, one could design a grading scheme by selecting some grading function and then awarding a student points whenever they indicate the correct answer with a confidence of . For instance, if the student was confident that the answer was “true” (and hence confident that the answer was “false”), then this grading scheme would award the student points if the correct answer actually was “true”, and points if the correct answer actually was “false”. One can then ask the question of what functions would be “best” for this scheme?

Intuitively, one would expect that should be monotone increasing – one should be rewarded more for being correct with high confidence, than correct with low confidence. On the other hand, some sort of “partial credit” should still be assigned in the latter case. One obvious proposal is to just use a linear grading function – thus for instance a correct answer given with confidence might be worth points. But is this the “best” option?

To make the problem more mathematically precise, one needs an objective criterion with which to evaluate a given grading scheme. One criterion that one could use here is the avoidance of perverse incentives. If a grading scheme is designed badly, a student may end up overstating or understating his or her confidence in an answer in order to optimise the (expected) grade: the optimal level of confidence for a student to report on a question may differ from that student’s subjective confidence . So one could ask to design a scheme so that is always equal to , so that the incentive is for the student to honestly report his or her confidence level in the answer.

This turns out to give a precise constraint on the grading function . If a student thinks that the answer to a question is true with probability and false with probability , and enters in an answer of “true” with confidence (and thus “false” with confidence ), then student would expect a grade of

on average for this question. To maximise this expected grade (assuming differentiability of , which is a reasonable hypothesis for a partial credit grading scheme), one performs the usual maneuvre of differentiating in the independent variable and setting the result to zero, thus obtaining

In order to avoid perverse incentives, the maximum should occur at , thus we should have

for all . This suggests that the function should be constant. (Strictly speaking, it only gives the weaker constraint that is symmetric around ; but if one generalised the problem to allow for multiple-choice questions with more than two possible answers, with a grading scheme that depended only on the confidence assigned to the correct answer, the same analysis would in fact force to be constant in ; we leave this computation to the interested reader.) In other words, should be of the form for some ; by monotonicity we expect to be positive. If we make the normalisation (so that no points are awarded for a split in confidence between true and false) and , one arrives at the grading scheme

Thus, if a student believes that an answer is “true” with confidence and “false” with confidence , he or she will be awarded points when the correct answer is “true”, and points if the correct answer is “false”. The following table gives some illustrative values for this scheme:

Confidence that answer is “true” | Points awarded if answer is “true” | Points awarded if answer is “false” |

Note the large penalties for being extremely confident of an answer that ultimately turns out to be incorrect; in particular, answers of confidence should be avoided unless one really is absolutely certain as to the correctness of one’s answer.

The total grade given under such a scheme to a student who answers each question to be “true” with confidence , and “false” with confidence , is

This grade can also be written as

where

is the likelihood of the student ‘s subjective probability model, given the outcome of the correct answers. Thus the grade system here has another natural interpretation, as being an affine rescaling of the log-likelihood. The incentive is thus for the student to maximise the likelihood of his or her own subjective model, which aligns well with standard practices in statistics. From the perspective of Bayesian probability, the grade given to a student can then be viewed as a measurement (in logarithmic scale) of how much the posterior probability that the student’s model was correct has improved over the prior probability.

One could propose using the above grading scheme to evaluate predictions to binary events, such as an upcoming election with only two viable candidates, to see in hindsight just how effective each predictor was in calling these events. One difficulty in doing so is that many predictions do not come with explicit probabilities attached to them, and attaching a default confidence level of to any prediction made without any such qualification would result in an automatic grade of if even one of these predictions turned out to be incorrect. But perhaps if a predictor refuses to attach confidence level to his or her predictions, one can assign some default level of confidence to these predictions, and then (using some suitable set of predictions from this predictor as “training data”) find the value of that maximises this predictor’s grade. This level can then be used going forward as the default level of confidence to apply to any future predictions from this predictor.

The above grading scheme extends easily enough to multiple-choice questions. But one question I had trouble with was how to deal with *uncertainty*, in which the student does not know enough about a question to venture even a probability of being true or false. Here, it is natural to allow a student to leave a question blank (i.e. to answer “I don’t know”); a more advanced option would be to allow the student to enter his or her confidence level as an interval range (e.g. “I am between and confident that the answer is “true””). But now I do not have a good proposal for a grading scheme; once there is uncertainty in the student’s subjective model, the problem of that student maximising his or her expected grade becomes ill-posed due to the “unknown unknowns”, and so the previous criterion of avoiding perverse incentives becomes far less useful.

When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology. However, there are some nice things one can do in an electronic medium, such as this blog. Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.

In the previous set of notes we established the central limit theorem, which we formulate here as follows:

Theorem 1 (Central limit theorem)Let be iid copies of a real random variable of mean and variance , and write . Then, for any fixed , we have

This is however not the end of the matter; there are many variants, refinements, and generalisations of the central limit theorem, and the purpose of this set of notes is to present a small sample of these variants.

First of all, the above theorem does not quantify the *rate* of convergence in (1). We have already addressed this issue to some extent with the Berry-Esséen theorem, which roughly speaking gives a convergence rate of uniformly in if we assume that has finite third moment. However there are still some quantitative versions of (1) which are not addressed by the Berry-Esséen theorem. For instance one may be interested in bounding the *large deviation probabilities*

in the setting where grows with . Chebyshev’s inequality gives an upper bound of for this quantity, but one can often do much better than this in practice. For instance, the central limit theorem (1) suggests that this probability should be bounded by something like ; however, this theorem only kicks in when is very large compared with . For instance, if one uses the Berry-Esséen theorem, one would need as large as or so to reach the desired bound of , even under the assumption of finite third moment. Basically, the issue is that convergence-in-distribution results, such as the central limit theorem, only really control the *typical* behaviour of statistics in ; they are much less effective at controlling the very rare *outlier* events in which the statistic strays far from its typical behaviour. Fortunately, there are large deviation inequalities (or *concentration of measure inequalities*) that do provide exponential type bounds for quantities such as (2), which are valid for both small and large values of . A basic example of this is the Chernoff bound that made an appearance in Exercise 47 of Notes 4; here we give some further basic inequalities of this type, including versions of the Bennett and Hoeffding inequalities.

In the other direction, we can also look at the fine scale behaviour of the sums by trying to control probabilities such as

where is now bounded (but can grow with ). The central limit theorem predicts that this quantity should be roughly , but even if one is able to invoke the Berry-Esséen theorem, one cannot quite see this main term because it is dominated by the error term in Berry-Esséen. There is good reason for this: if for instance takes integer values, then also takes integer values, and can vanish when is less than and is slightly larger than an integer. However, this turns out to essentially be the only obstruction; if does not lie in a lattice such as , then we can establish a *local limit theorem* controlling (3), and when does take values in a lattice like , there is a discrete local limit theorem that controls probabilities such as . Both of these limit theorems will be proven by the Fourier-analytic method used in the previous set of notes.

We also discuss other limit theorems in which the limiting distribution is something other than the normal distribution. Perhaps the most common example of these theorems is the Poisson limit theorems, in which one sums a large number of indicator variables (or approximate indicator variables), each of which is rarely non-zero, but which collectively add up to a random variable of medium-sized mean. In this case, it turns out that the limiting distribution should be a Poisson random variable; this again is an easy application of the Fourier method. Finally, we briefly discuss limit theorems for other stable laws than the normal distribution, which are suitable for summing random variables of infinite variance, such as the Cauchy distribution.

Finally, we mention a very important class of generalisations to the CLT (and to the variants of the CLT discussed in this post), in which the hypothesis of joint independence between the variables is relaxed, for instance one could assume only that the form a martingale. Many (though not all) of the proofs of the CLT extend to these more general settings, and this turns out to be important for many applications in which one does not expect joint independence. However, we will not discuss these generalisations in this course, as they are better suited for subsequent courses in this series when the theory of martingales, conditional expectation, and related tools are developed.

Let be iid copies of an absolutely integrable real scalar random variable , and form the partial sums . As we saw in the last set of notes, the law of large numbers ensures that the empirical averages converge (both in probability and almost surely) to a deterministic limit, namely the mean of the reference variable . Furthermore, under some additional moment hypotheses on the underlying variable , we can obtain *square root cancellation* for the fluctuation of the empirical average from the mean. To simplify the calculations, let us first restrict to the case of mean zero and variance one, thus

and

Then, as computed in previous notes, the normalised fluctuation also has mean zero and variance one:

This and Chebyshev’s inequality already indicates that the “typical” size of is , thus for instance goes to zero in probability for any that goes to infinity as . If we also have a finite fourth moment , then the calculations of the previous notes also give a fourth moment estimate

From this and the Paley-Zygmund inequality (Exercise 42 of Notes 1) we also get some lower bound for of the form

for some absolute constant and for sufficiently large; this indicates in particular that does not converge in any reasonable sense to something finite for any that goes to infinity.

The question remains as to what happens to the ratio itself, without multiplying or dividing by any factor . A first guess would be that these ratios converge in probability or almost surely, but this is unfortunately not the case:

Proposition 1Let be iid copies of an absolutely integrable real scalar random variable with mean zero, variance one, and finite fourth moment, and write . Then the random variables do not converge in probability or almost surely to any limit, and neither does any subsequence of these random variables.

*Proof:* Suppose for contradiction that some sequence converged in probability or almost surely to a limit . By passing to a further subsequence we may assume that the convergence is in the almost sure sense. Since all of the have mean zero, variance one, and bounded fourth moment, Theorem 24 of Notes 1 implies that the limit also has mean zero and variance one. On the other hand, is a tail random variable and is thus almost surely constant by the Kolmogorov zero-one law from Notes 3. Since constants have variance zero, we obtain the required contradiction.

Nevertheless there is an important limit for the ratio , which requires one to replace the notions of convergence in probability or almost sure convergence by the weaker concept of convergence in distribution.

Definition 2 (Vague convergence and convergence in distribution)Let be a locally compact Hausdorff topological space with the Borel -algebra. A sequence of finite measures on is said to converge vaguely to another finite measure if one hasas for all continuous compactly supported functions . (Vague convergence is also known as

weak convergence, although strictly speaking the terminology weak-* convergence would be more accurate.) A sequence of random variables taking values in is said toconverge in distribution(orconverge weaklyorconverge in law) to another random variable if the distributions converge vaguely to the distribution , or equivalently ifas for all continuous compactly supported functions .

One could in principle try to extend this definition beyond the locally compact Hausdorff setting, but certain pathologies can occur when doing so (e.g. failure of the Riesz representation theorem), and we will never need to consider vague convergence in spaces that are not locally compact Hausdorff, so we restrict to this setting for simplicity.

Note that the notion of convergence in distribution depends only on the distribution of the random variables involved. One consequence of this is that convergence in distribution does not produce unique limits: if converges in distribution to , and has the same distribution as , then also converges in distribution to . However, limits are unique up to equivalence in distribution (this is a consequence of the Riesz representation theorem, discussed for instance in this blog post). As a consequence of the insensitivity of convergence in distribution to equivalence in distribution, we may also legitimately talk about convergence of distribution of a sequence of random variables to another random variable even when all the random variables and involved are being modeled by different probability spaces (e.g. each is modeled by , and is modeled by , with no coupling presumed between these spaces). This is in contrast to the stronger notions of convergence in probability or almost sure convergence, which require all the random variables to be modeled by a common probability space. Also, by an abuse of notation, we can say that a sequence of random variables converges in distribution to a probability measure , when converges vaguely to . Thus we can talk about a sequence of random variables converging in distribution to a uniform distribution, a gaussian distribution, etc..

From the dominated convergence theorem (available for both convergence in probability and almost sure convergence) we see that convergence in probability or almost sure convergence implies convergence in distribution. The converse is not true, due to the insensitivity of convergence in distribution to equivalence in distribution; for instance, if are iid copies of a non-deterministic scalar random variable , then the trivially converge in distribution to , but will not converge in probability or almost surely (as one can see from the zero-one law). However, there are some partial converses that relate convergence in distribution to convergence in probability; see Exercise 10 below.

Remark 3The notion of convergence in distribution is somewhat similar to the notion of convergence in the sense of distributions that arises in distribution theory (discussed for instance in this previous blog post), however strictly speaking the two notions of convergence are distinct and should not be confused with each other, despite the very similar names.

The notion of convergence in distribution simplifies in the case of real scalar random variables:

Proposition 4Let be a sequence of scalar random variables, and let be another scalar random variable. Then the following are equivalent:

- (i) converges in distribution to .
- (ii) converges to for each continuity point of (i.e. for all real numbers at which is continuous). Here is the cumulative distribution function of .

*Proof:* First suppose that converges in distribution to , and is continuous at . For any , one can find a such that

for every . One can also find an larger than such that and . Thus

and

Let be a continuous function supported on that equals on . Then by the above discussion we have

and hence

for large enough . In particular

A similar argument, replacing with a continuous function supported on that equals on gives

for large enough. Putting the two estimates together gives

for large enough; sending , we obtain the claim.

Conversely, suppose that converges to at every continuity point of . Let be a continuous compactly supported function, then it is uniformly continuous. As is monotone increasing, it can only have countably many points of discontinuity. From these two facts one can find, for any , a simple function for some that are points of continuity of , and real numbers , such that for all . Thus

Similarly for replaced by . Subtracting and taking limit superior, we conclude that

and on sending , we obtain that converges in distribution to as claimed.

The restriction to continuity points of is necessary. Consider for instance the deterministic random variables , then converges almost surely (and hence in distribution) to , but does not converge to .

Example 5For any natural number , let be a discrete random variable drawn uniformly from the finite set , and let be the continuous random variable drawn uniformly from . Then converges in distribution to . Thus we see that a continuous random variable can emerge as the limit of discrete random variables.

Example 6For any natural number , let be a continuous random variable drawn uniformly from , then converges in distribution to the deterministic real number . Thus we see that discrete (or even deterministic) random variables can emerge as the limit of continuous random variables.

Exercise 7 (Portmanteau theorem)Show that the properties (i) and (ii) in Proposition 4 are also equivalent to the following three statements:

- (iii) One has for all closed sets .
- (iv) One has for all open sets .
- (v) For any Borel set whose topological boundary is such that , one has .
(Note: to prove this theorem, you may wish to invoke Urysohn’s lemma. To deduce (iii) from (i), you may wish to start with the case of compact .)

We can now state the famous central limit theorem:

Theorem 8 (Central limit theorem)Let be iid copies of a scalar random variable of finite mean and finite non-zero variance . Let . Then the random variables converges in distribution to a random variable with the standard normal distribution (that is to say, a random variable with probability density function ). Thus, by abuse of notationIn the normalised case when has mean zero and unit variance, this simplifies to

Using Proposition 4 (and the fact that the cumulative distribution function associated to is continuous, the central limit theorem is equivalent to asserting that

as for any , or equivalently that

Informally, one can think of the central limit theorem as asserting that approximately behaves like it has distribution for large , where is the normal distribution with mean and variance , that is to say the distribution with probability density function . The integrals can be written in terms of the error function as .

The central limit theorem is a basic example of the *universality phenomenon* in probability – many statistics involving a large system of many independent (or weakly dependent) variables (such as the normalised sums ) end up having a universal asymptotic limit (in this case, the normal distribution), regardless of the precise makeup of the underlying random variable that comprised that system. Indeed, the universality of the normal distribution is such that it arises in many other contexts than the fluctuation of iid random variables; the central limit theorem is merely the first place in probability theory where it makes a prominent appearance.

We will give several proofs of the central limit theorem in these notes; each of these proofs has their advantages and disadvantages, and can each extend to prove many further results beyond the central limit theorem. We first give Lindeberg’s proof of the central limit theorem, based on exchanging (or swapping) each component of the sum in turn. This proof gives an accessible explanation as to why there should be a universal limit for the central limit theorem; one then computes directly with gaussians to verify that it is the normal distribution which is the universal limit. Our second proof is the most popular one taught in probability texts, namely the Fourier-analytic proof based around the concept of the characteristic function of a real random variable . Thanks to the powerful identities and other results of Fourier analysis, this gives a quite short and direct proof of the central limit theorem, although the arguments may seem rather magical to readers who are not already familiar with Fourier methods. Finally, we give a proof based on the moment method, in the spirit of the arguments in the previous notes; this argument is more combinatorial, but is straightforward and is particularly robust, in particular being well equipped to handle some dependencies between components; we will illustrate this by proving the Erdos-Kac law in number theory by this method. Some further discussion of the central limit theorem (including some further proofs, such as one based on Stein’s method) can be found in this blog post. Some further variants of the central limit theorem, such as local limit theorems, stable laws, and large deviation inequalities, will be discussed in the next (and final) set of notes.

The following exercise illustrates the power of the central limit theorem, by establishing combinatorial estimates which would otherwise require the use of Stirling’s formula to establish.

Exercise 9 (De Moivre-Laplace theorem)Let be a Bernoulli random variable, taking values in with , thus has mean and variance . Let be iid copies of , and write .

- (i) Show that takes values in with . (This is an example of a binomial distribution.)
- (ii) Assume Stirling’s formula
where is a function of that goes to zero as . (A proof of this formula may be found in this previous blog post.) Using this formula, and without using the central limit theorem, show that

as for any fixed real numbers .

The above special case of the central limit theorem was first established by de Moivre and Laplace.

We close this section with some basic facts about convergence of distribution that will be useful in the sequel.

Exercise 10Let , be sequences of real random variables, and let be further real random variables.

- (i) If is deterministic, show that converges in distribution to if and only if converges in probability to .
- (ii) Suppose that is independent of for each , and independent of . Show that converges in distribution to if and only if converges in distribution to and converges in distribution to . (The shortest way to prove this is by invoking the Stone-Weierstrass theorem, but one can also proceed by proving some version of Proposition 4.) What happens if the independence hypothesis is dropped?
- (iii) If converges in distribution to , show that for every there exists such that for all sufficiently large . (That is to say, is a tight sequence of random variables.)
- (iv) Show that converges in distribution to if and only if, after extending the probability space model if necessary, one can find copies and of and respectively such that converges almost surely to . (
Hint:use the Skorohod representation, Exercise 29 of Notes 0.)- (v) If converges in distribution to , and is continuous, show that converges in distribution to . Generalise this claim to the case when takes values in an arbitrary locally compact Hausdorff space.
- (vi) (Slutsky’s theorem) If converges in distribution to , and converges in probability to a
deterministiclimit , show that converges in distribution to , and converges in distribution to . (Hint: either use (iv), or else use (iii) to control some error terms.) This statement combines particularly well with (i). What happens if is not assumed to be deterministic?- (vii) (Fatou lemma) If is continuous, and converges in distribution to , show that .
- (viii) (Bounded convergence) If is continuous and bounded, and converges in distribution to , show that .
- (ix) (Dominated convergence) If converges in distribution to , and there is an absolutely integrable such that almost surely for all , show that .

For future reference we also mention (but will not prove) Prokhorov’s theorem that gives a partial converse to part (iii) of the above exercise:

Theorem 11 (Prokhorov’s theorem)Let be a sequence of real random variables which is tight (that is, for every there exists such that for all sufficiently large ). Then there exists a subsequence which converges in distribution to some random variable (which may possibly be modeled by a different probability space model than the .)

The proof of this theorem relies on the Riesz representation theorem, and is beyond the scope of this course; but see for instance Exercise 29 of this previous blog post. (See also the closely related Helly selection theorem, covered in Exercise 30 of the same post.)

One of the major activities in probability theory is studying the various statistics that can be produced from a complex system with many components. One of the simplest possible systems one can consider is a finite sequence or an infinite sequence of jointly independent scalar random variables, with the case when the are also identically distributed (i.e. the are iid) being a model case of particular interest. (In some cases one may consider a triangular array of scalar random variables, rather than a finite or infinite sequence.) There are many statistics of such sequences that one can study, but one of the most basic such statistics are the partial sums

The first fundamental result about these sums is the law of large numbers (or LLN for short), which comes in two formulations, weak (WLLN) and strong (SLLN). To state these laws, we first must define the notion of convergence in probability.

Definition 1Let be a sequence of random variables taking values in a separable metric space (e.g. the could be scalar random variables, taking values in or ), and let be another random variable taking values in . We say that converges in probability to if, for every radius , one has as . Thus, if are scalar, we have converging to in probability if as for any given .

The measure-theoretic analogue of convergence in probability is convergence in measure.

It is instructive to compare the notion of convergence in probability with almost sure convergence. it is easy to see that converges almost surely to if and only if, for every radius , one has as ; thus, roughly speaking, convergence in probability is good for controlling how a single random variable is close to its putative limiting value , while almost sure convergence is good for controlling how the entire *tail* of a sequence of random variables is close to its putative limit .

We have the following easy relationships between convergence in probability and almost sure convergence:

Exercise 2Let be a sequence of scalar random variables, and let be another scalar random variable.

- (i) If almost surely, show that in probability. Give a counterexample to show that the converse does not necessarily hold.
- (ii) Suppose that for all . Show that almost surely. Give a counterexample to show that the converse does not necessarily hold.
- (iii) If in probability, show that there is a subsequence of the such that almost surely.
- (iv) If are absolutely integrable and as , show that in probability. Give a counterexample to show that the converse does not necessarily hold.
- (v) (Urysohn subsequence principle) Suppose that every subsequence of has a further subsequence that converges to in probability. Show that also converges to in probability.
- (vi) Does the Urysohn subsequence principle still hold if “in probability” is replaced with “almost surely” throughout?
- (vii) If converges in probability to , and or is continuous, show that converges in probability to . More generally, if for each , is a sequence of scalar random variables that converge in probability to , and or is continuous, show that converges in probability to . (Thus, for instance, if and converge in probability to and respectively, then and converge in probability to and respectively.
- (viii) (Fatou’s lemma for convergence in probability) If are non-negative and converge in probability to , show that .
- (ix) (Dominated convergence in probability) If converge in probability to , and one almost surely has for all and some absolutely integrable , show that converges to .

Exercise 3Let be a sequence of scalar random variables converging in probability to another random variable .

- (i) Suppose that there is a random variable which is independent of for each individual . Show that is also independent of .
- (ii) Suppose that the are jointly independent. Show that is almost surely constant (i.e. there is a deterministic scalar such that almost surely).

We can now state the weak and strong law of large numbers, in the model case of iid random variables.

Theorem 4 (Law of large numbers, model case)Let be an iid sequence of copies of an absolutely integrable random variable (thus the are independent and all have the same distribution as ). Write , and for each natural number , let denote the random variable .

- (i) (Weak law of large numbers) The random variables converge in probability to .
- (ii) (Strong law of large numbers) The random variables converge almost surely to .

Informally: if are iid with mean , then for large. Clearly the strong law of large numbers implies the weak law, but the weak law is easier to prove (and has somewhat better quantitative estimates). There are several variants of the law of large numbers, for instance when one drops the hypothesis of identical distribution, or when the random variable is not absolutely integrable, or if one seeks more quantitative bounds on the rate of convergence; we will discuss some of these variants below the fold.

It is instructive to compare the law of large numbers with what one can obtain from the Kolmogorov zero-one law, discussed in Notes 2. Observe that if the are real-valued, then the limit superior and are tail random variables in the sense that they are not affected if one changes finitely many of the ; in particular, events such as are tail events for any . From this and the zero-one law we see that there must exist deterministic quantities such that and almost surely. The strong law of large numbers can then be viewed as the assertion that when is absolutely integrable. On the other hand, the zero-one law argument does not require absolute integrability (and one can replace the denominator by other functions of that go to infinity as ).

The law of large numbers asserts, roughly speaking, that the theoretical expectation of a random variable can be approximated by taking a large number of independent samples of and then forming the empirical mean . This ability to approximate the theoretical statistics of a probability distribution through empirical data is one of the basic starting points for mathematical statistics, though this is not the focus of the course here. The tendency of statistics such as to cluster closely around their mean value is the simplest instance of the concentration of measure phenomenon, which is of tremendous significance not only within probability, but also in applications of probability to disciplines such as statistics, theoretical computer science, combinatorics, random matrix theory and high dimensional geometry. We will not discuss these topics much in this course, but see this previous blog post for some further discussion.

There are several ways to prove the law of large numbers (in both forms). One basic strategy is to use the *moment method* – controlling statistics such as by computing moments such as the mean , variance , or higher moments such as for . The joint independence of the make such moments fairly easy to compute, requiring only some elementary combinatorics. A direct application of the moment method typically requires one to make a finite moment assumption such as , but as we shall see, one can reduce fairly easily to this case by a truncation argument.

For the strong law of large numbers, one can also use methods relating to the theory of martingales, such as stopping time arguments and maximal inequalities; we present some classical arguments of Kolmogorov in this regard.

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