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 $\{X,X_n;n\ge 1\}$ be a sequence of independent and identically distributed Banach space valued random variables and set $S_n={\sum }_{i=1}^n{X}_i,n\ge 1$. Let $\{a_n;n\ge 1\}$ and $\{b_n;n\ge 1\}$ be increasing sequences of positive real numbers such that ${\lim }_{n\to \infty }{a}_n=\infty$ and $\left\{b_n/a_n;n\ge 1\right\}$ 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 ${\sum }_{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >b_n)<\infty$ if $S_n/a_n\to 0$ almost surely for every symmetric Banach space valued random variable $X$ with ${\sum }_{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >a_n)<\infty$. 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.

10 pages. arXiv admin note: text overlap with arXiv:1703.07868. text overlap with arXiv:1506.07596

References in corpus (1)

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