The Hsu–Robbins–Erdös theorem for the maximum partial sums of quadruplewise independent random variables

Abstract

Etemadi (1981) [10] and Rio (1995) [27] provided proofs of the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers under optimal moment conditions without using the Kolmogorov-type maximal inequalities. While this famous result holds for sequences of pairwise independent identically distributed real-valued random variables, a closely related result, the Hsu–Robbins–Erdös strong law of large numbers may fail if the underlying random variables are only assumed to be pairwise independent identically distributed. This note develops Rio's method and uses an approximation technique to establish the Hsu–Robbins–Erdös strong law of large numbers for the maximum partial sums of quadruplewise independent identically distributed random variables. We consider random variables taking values in a real separable Banach space X, but the main result is new even when X is the real line. Previous contributions so far considered the complete convergence of the partial sums or restricted to dependence structures satisfying a Kolmogorov-type maximal inequality.