On the central limit theorem along subsequences of sums of i.i.d. random variables

Abstract

Let $\mathbb{N } = \{1, 2, 3, \ldots \}$. Let $\{X, X_{n}; n \in \mathbb N \}$ be a sequence of i.i.d. random variables, and let $S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N$. Then $S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})$ for some $\sigma ^{2} < \infty$ whenever, for a subsequence $\{n_{k}; k \in \mathbb N \}$ of $\mathbb N$, $S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})$. Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when $\{\sqrt{n}; n \in \mathbb N \}$ is replaced by $\{\sqrt{na_{n}};n \in \mathbb N \}$ with $\lim _{n \rightarrow \infty } a_{n} = \infty$. We show that, for given positive nondecreasing sequence $\{a_{n}; n \in \mathbb N \}$ with $\lim _{n \rightarrow \infty } a_{n} = \infty$ and $\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1$ and given nondecreasing function $h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$ with $\lim _{x \rightarrow \infty } h(x) = \infty$, there exists a sequence $\{X, X_{n}; n \in \mathbb N \}$ of symmetric i.i.d. random variables such that $\mathbb E h(|X|) = \infty$ and, for some subsequence $\{n_{k}; k \in \mathbb N \}$ of $\mathbb N$, $S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)$. In particular, for given $0 < p < 2$ and given nondecreasing function $h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$ with $\lim _{x \rightarrow \infty } h(x) = \infty$, there exists a sequence $\{X, X_{n}; n \in \mathbb N \}$ of symmetric i.i.d. random variables such that $\mathbb E h(|X|) = \infty$ and, for some subsequence $\{n_{k}; k \in \mathbb N \}$ of $\mathbb N$, $S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)$.