A comparison theorem for the law of large numbers in Banach spaces

TL;DR

This paper establishes a comparison theorem for the law of large numbers in Banach spaces, linking convergence properties of sums of i.i.d. Banach space-valued variables under different normalization sequences.

Contribution

It introduces a novel comparison theorem for the law of large numbers in Banach spaces, utilizing symmetrization and sum comparison tools.

Findings

Provides conditions under which normalized sums converge almost surely or in probability.

Connects convergence behaviors under different scaling sequences.

Offers applications and consequences of the main comparison theorem.

Abstract

Let $(\mathbf{B}, \|\cdot\|)$ be a real separable Banach space. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. {\bf B}-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let $\{a_{n}; n \geq 1\}$ and $\{b_{n}; n \geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left\{b_{n}/a_{n};~ n \geq 1 \right\}$ is a nondecreasing sequence. In this paper, we provide a comparison theorem for the law of large numbers for i.i.d. {\bf B}-valued random variables. That is, we show that $\displaystyle \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0$ almost surely (resp. in probability) for every {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > b_{n}) = 0$) if $S_{n}/a_{n} \rightarrow 0$ almost surely (resp. in probability) for every symmetric {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > a_{n}) = 0$). To establish this comparison theorem for the law of large numbers, we invoke two tools: 1) a comparison theorem for sums of independent {\bf B}-valued random variables and, 2) a symmetrization procedure for the law of large numbers for sums of independent {\bf B}-valued random variables. A few consequences of our main results are provided.