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 1** Let 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 has

as 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 to *converge in distribution* (or *converge weakly* or *converge in law*) to another random variable if the distributions converge vaguely to the distribution , or equivalently if

as 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 3** The 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 4** Let 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 5** For 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 6** For 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 notation

In 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 .

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 10** Let , 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
*deterministic* limit , 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.)

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