~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.
Categories: math.PR