تقرير
A comparison theorem for the law of large numbers in Banach spaces
| العنوان: | A comparison theorem for the law of large numbers in Banach spaces |
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| المؤلفون: | Li, Deli, Liang, Han-Ying |
| سنة النشر: | 2015 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, 60F15, 60F05, 60B12, 60G50 |
| الوصف: | 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. Comment: 19 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1506.07596 |
| رقم الأكسشن: | edsarx.1506.07596 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |