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.