A probability inequality for sums of independent Banach space valued random variables

Abstract

Let $(\mathbf{B},\Vert \cdot \Vert )$ be a real separable Banach space. Let $\phi (\cdot )$ and $\psi (\cdot )$ be two continuous and increasing functions defined on $[0,\infty )$ such that $\phi (0)=\psi (0)=0$, ${\lim }_{t\to \infty }\phi (t)=\infty$, and $\frac{\psi (\cdot )}{\phi (\cdot )}$ is a nondecreasing function on $[0,\infty )$. Let $\{V_n;n\ge 1\}$ 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 $n\ge 1$ and all $t\ge 0$, \[ \mathbb{P}\left(\left\|\sum_{i=1}^{n} V_{i} \right\| > t b_{n} \right) \leq 4 \mathbb{P} \left(\left\|\sum_{i=1}^{n} φ\left(ψ^{-1}(\|V_{i}\|)\right) \frac{V_{i}}{\|V_{i}\|} \right\| > t a_{n} \right) + \sum_{i=1}^{n}\mathbb{P}\left(\|V_{i}\| > b_{n} \right), \] where $a_n=\phi (n)$ and $b_n=\psi (n)$, $n\ge 1$. 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 $\mathbf{B}$-valued random variables.

10 pages. arXiv admin note: substantial text overlap with arXiv:1506.07596

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