Some Results on the Central Limit Theorem for Subsequences in Banach Spaces

TL;DR

This paper investigates the weak convergence of normalized sums of i.i.d. Banach space-valued random variables, establishing conditions under which subsequence convergence implies full sequence convergence in spaces of cotype 2.

Contribution

It proves that in cotype 2 Banach spaces, subsequence weak convergence of normalized sums implies full sequence convergence, and explores implications for non-convergent cases.

Findings

Weak convergence of normalized sums is equivalent for the full sequence and subsequences in cotype 2 spaces.

If subsequence converges but the full sequence does not, no normalization yields a non-degenerate limit.

Conjecture that the equivalence fails in spaces not of cotype 2.

Abstract

Let $\{X, X_{n}; n \geq 1 \}$ be a sequence of i.i.d. $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. This note is devoted to study the classical central limitr theorem for subsequences of sums of i.i.d. $\mathbf{B}$-valued random variables. We show that, under the assumption that $\mathbf{B}$ is of cotype $2$ space, $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ converges weakly if and only if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. We conjecture that this result is false if $\mathbf{B}$ is not of cotype $2$ space. In addition, we show that, if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. and $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ does not converge weakly, then $\displaystyle \left(S_{n}/a_{n} \right)_{n \geq 1}$ does not converge weakly to a non-degenerate probability measure for any sequence $\{a_{n}; n \geq 1 \}$ of positive real numbers.