تقرير
A probability inequality for sums of independent Banach space valued random variables
| العنوان: | A probability inequality for sums of independent Banach space valued random variables |
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| المؤلفون: | Li, Deli, Liang, Han-Ying, Rosalsky, Andrew |
| سنة النشر: | 2017 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, 60E15, 60B12, 60G50 |
| الوصف: | Let $(\mathbf{B}, \|\cdot\|)$ be a real separable Banach space. Let $\varphi(\cdot)$ and $\psi(\cdot)$ be two continuous and increasing functions defined on $[0, \infty)$ such that $\varphi(0) = \psi(0) = 0$, $\lim_{t \rightarrow \infty} \varphi(t) = \infty$, and $\frac{\psi(\cdot)}{\varphi(\cdot)}$ is a nondecreasing function on $[0, \infty)$. Let $\{V_{n};~n \geq 1 \}$ be a sequence of independent and symmetric {\bf B}-valued random variables. In this note, we establish a probability inequality for sums of independent {\bf B}-valued random variables by showing that for every $n \geq 1$ and all $t \geq 0$, \[ \mathbb{P}\left(\left\|\sum_{i=1}^{n} V_{i} \right\| > t b_{n} \right) \leq 4 \mathbb{P} \left(\left\|\sum_{i=1}^{n} \varphi\left(\psi^{-1}(\|V_{i}\|)\right) \frac{V_{i}}{\|V_{i}\|} \right\| > t a_{n} \right) + \sum_{i=1}^{n}\mathbb{P}\left(\|V_{i}\| > b_{n} \right), \] where $a_{n} = \varphi(n)$ and $b_{n} = \psi(n)$, $n \geq 1$. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed ${\bf B}$-valued random variables. Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1506.07596 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1703.07868 |
| رقم الأكسشن: | edsarx.1703.07868 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |