A comparison theorem for the law of large numbers in Banach spaces

Let $(\mathbf{B},\mathrm{\parallel }\cdot \mathrm{\parallel })$ be a real separable Banach space. Let ${X, X_{n}; n \geq 1}$ be a sequence of i.i.d. {\bf B}-valued random variables and set $S_n={\sum }_{i=1}^n{X}_i,n\ge 1$. Let $\{a_n;n\ge 1\}$ and $\{b_n;n\ge 1\}$ be increasing sequences of positive real numbers such that ${\lim }_{n\to \mathrm{\infty }}{a}_n=\mathrm{\infty }$ and $\left{b_{n}/a_{n};~ n \geq 1 \right}$ is a nondecreasing sequence. In this paper, we provide a comparison theorem for the law of large numbers for i.i.d. {\bf B}-valued random variables. That is, we show that $\displaystyle \frac{S_{n}- n $\mathbb{E}$\left(XI{|X| \leq b_{n} } \right)}{b_{n}} \rightarrow 0$ almost surely (resp. in probability) for every {\bf B}-valued random variable $X$ with ${\sum }_{n=1}^{\mathrm{\infty }}\mathbb{P}(\mathrm{\parallel }X\mathrm{\parallel }>b_n)<\mathrm{\infty }$ (resp. $\lim_{n \rightarrow \infty}n$\mathbb{P}$(|X| > b_{n}) = 0$)if$S_{n}/a_{n} \rightarrow 0$ almost surely (resp. in probability) for every symmetric {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} $\mathbb{P}$(|X| > a_{n}) < \infty$(resp\mathrm{.}$\lim_{n \rightarrow \infty}n$\mathbb{P}$(|X| > a_{n}) = 0$). To establish this comparison theorem for the law of large numbers, we invoke two tools: 1) a comparison theorem for sums of independent {\bf B}-valued random variables and, 2) a symmetrization procedure for the law of large numbers for sums of independent {\bf B}-valued random variables. A few consequences of our main results are provided.