arXiv 1506.075962015-06-25math.PR

巴拿赫空间中law of large numbers的一个比较定理

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

Deli Li, Han-Ying Liang

备注：: 19 pages

摘要：

设$(\mathbf{B}, \|\cdot\|)$是一个实可分Banach空间。令$\{X, X_{n}; n \geq 1\}$是i.i.d. { \bf B}值随机变量和集合$S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$。设$\{a_{n}; n \geq 1\}$和$\{b_{n}; n \geq 1\}$是正实数的递增序列，使得$\lim_{n \rightarrow \infty} a_{n} = \infty$和$\left\{b_{n}/a_{n};~ n \geq 1 \right\}$是非减序列.在本文中，我们提供了一个比较定理的大数定律i.i.d. { \bf B}值随机变量。也就是说，我们表明，$\displaystyle \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0$几乎肯定（分别）。对于每个{ \bf B}值随机变量$X$，其中$\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$（分别为$\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > b_{n}) = 0$），如果$S_{n}/a_{n} \rightarrow 0$几乎肯定（分别）。对于每个对称{ \bf B}值随机变量$X$，其中$\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$（分别为$\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > a_{n}) = 0$）。为了建立这个大数定律的比较定理，我们调用两个工具：1）独立{ \bf B}值随机变量和的比较定理，2）独立{ \bf B}值随机变量和的大数定律的对称化过程。我们的主要结果的一些后果。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文