Some Exponential Moments of Sums of Independent Random Variables

Abstract

If $\{ {X_n}\}$ is a sequence of vector valued random variables, $\{ {a_n}\}$ a sequence of positive constants, and $M = {\sup _{n \geqslant 1}}\left \| {({X_1} + \cdots + {X_n})/{a_n}} \right \|$, we examine when $E(\Phi (M)) < \infty$ under various conditions on $\Phi ,\{ {X_n}\}$, and $\{ {a_n}\}$. These integrability results easily apply to empirical distribution functions.