An Extension of Feller's Strong Law of Large Numbers

Abstract: ~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let ${X, X_{n}; n \geq 1}$ be a sequence of independent and identically distributed Banach space valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let ${a_{n}; n \geq 1}$ and ${b_{n}; n \geq 1}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left{b_{n}/a_{n};~ n \geq 1 \right}$ is a nondecreasing sequence. We show that [ \frac{S_{n}- n \mathbb{E}\left(XI{|X| \leq b_{n} } \right)}{b_{n}} \rightarrow 0~~\mbox{almost surely} ] for every Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(|X| > b_{n}) < \infty$ if $S_{n}/a_{n} \rightarrow 0$ almost surely for every symmetric Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(|X| > a_{n}) < \infty$. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.