Petrov's CLT template

In his 1975 textbook Sums of independent random variables Petrov proved a number of useful CLT templates. Here are two extremely useful ones.

Theorem 15

Let $\{X_{k,n}\}$, $1\le k\le n$, $n=1,2,\dots$ be a set of sequences of random variables, where the random variables in each sequence are independent. Then

$$\frac{1}{n}\sum _{k\le n}{X}_{k,n}\to N(\mu ,{\sigma }^2),$$

if and only if for all $ϵ>0$ the following three conditions are all satisfied:

• ${\sum }_{k\le n}\mathbb{P}(|X_{n,k}|\ge ϵ)\to 0,$
• ${\sum }_{k\le n}\mathbb{V}(X_{k,n}1(|X_{k,n}|<ϵ))\to {\sigma }^2$
• ${\sum }_{k\le n}\mathbb{E}[X_{k,n}1(|X_{k,n}|<ϵ)]\to \mu$.

Note that this strengthens the Lindeberg-Feller CLT (and others) as these are necessary and sufficient conditions.

Theorem 21

Let $\{X_n\}$ be a sequence of independent random variables. Then

$${\rho }_{\text{KS}}\left(\frac{1}{a_n}\sum _{k\le n}{X}_k-b_n,N(0,1)\right)\to 0,$$

and ${\max }_{k\le n}\mathbb{P}(|X_k|\ge ϵa_n)\to 0$ if and only if there exists some sequence $(c_n)$ with $c_n\to \infty$ and the following two conditions hold:

• ${\sum }_{k\le n}\mathbb{P}(|X_k|\le c_n)\to 0$,
• $\frac{1}{c_n^2}{\sum }_{k\le n}\mathbb{V}(X_{k}1(|X_k|<c_n))\to \infty$. In this case we can take

$$a_n^2=\sum _{k\le n}\mathbb{V}(X_{k}1(|X_k|<c_n)),b_n=\frac{1}{a_n}\sum _{k\le n}\mathbb{E}[X_{k}1|X|<c_n].$$

Petrov also show that the condition ${\sum }_k\mathbb{P}(|X_{k,n}|\ge ϵ)\to 0$ used in Theorem 15 is equivalent to $\mathbb{P}({\max }_k|X_{k,n}|\ge ϵ)\to 0$, which is stronger than what is used in Theorem 21. But intuitively, this is where these kind of “maximal” conditions come from. If such a condition doesn’t hold, then the random variables are becoming too large over time to converge nicely.

Theorem 19

Related to Theorem 21 but with slightly different conditions. Let $\{X_n\}$ be a sequence of independent random variables. Then

$${\rho }_{\text{KS}}\left(\frac{1}{a_n}\sum _{k\le n}{X}_k,N(0,1)\right)\to 0,$$

and ${\max }_{k\le n}\mathbb{P}(|X_k|\ge ϵa_n)\to 0$ if and only if

• ${\sum }_{k\le n}\mathbb{P}(|X_k|\ge ϵa_n)\to 0$ for all $ϵ>0$,
• $\frac{1}{a_n^2}{\sum }_{k\le n}\mathbb{V}(X_{k}1(|X_k|<a_n))\to 1$, and
• $\frac{1}{a_n}{\sum }_{k\le n}\mathbb{E}[X_{k}1(|X_k|<a_n)]\to 0$.

This is arguably more useful than Theorem 21 as we don’t have to find the sequence $c_n$; $a_n$ is often given.