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

Let $(\mathbf{B},\mathrm{\parallel }\cdot \mathrm{\parallel })$ be a real separable Banach space. Let $\phi (\cdot )$ and $\psi (\cdot )$ be two continuous and increasing functions defined on $[0,\mathrm{\infty })$ such that $\phi (0)=\psi (0)=0$, $\lim_{t \rightarrow \infty} \varphi(t) = \infty$,and$\frac{\psi(\cdot)}{\varphi(\cdot)}$isanondecreasingfunctionon$[0, \infty)$\mathrm{.}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 $n\ge 1$ and all $t\ge 0$, $$\mathbb{P}(\parallel \sum _{i=1}^n{V}_i\parallel >t{b}_n)\le 4\mathbb{P}(\parallel \sum _{i=1}^n\phi ({\psi }^{-1}(\mathrm{\parallel }{V}_i\mathrm{\parallel }))\frac{V_i}{\mathrm{\parallel }{V}_i\mathrm{\parallel }}\parallel >t{a}_n)+\sum _{i=1}^n\mathbb{P}(\mathrm{\parallel }{V}_i\mathrm{\parallel }>b_n),$$ 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 $\mathbf{B}$-valued random variables.