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

Abstract

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.