ABSTRACT
In this chapter we study1 the asymptotics of almost sure and tail behavior of sums, (Sn/n
1/p) = (X1 + · · · + Xn)/n1/p), 1 ≤ p < 2, for independent, centered random vectors Xn, n = 1, 2, . . . , and of martingales, (Mn), taking values in a Banach space X. The obtained results are in the spirit of classical theorems of Marcinkiewicz-Zugmund, Hsu-Robbins-Erdo¨s-Spitzer, and Brunk, for real-valued random variables, and show the essential role played by the geometry of X in the infinite-dimensional case.
