The Robbins-Siegmund Series Criterion for Partial Maxima

Abstract

Let $X, X_1, X_2,\cdots$ be i.i.d. random variables and let $M_n = \max_{1\leq j \leq n}X_j$. For each nondecreasing real sequence $\{b_n\}$ such that $P(X > b_n) \rightarrow 0$ and $P(M_n \leq b_n) \rightarrow 0$ we show that $P(M_n \leq b_n i.o.) = 1$ if and only if $\sum_nP(X > b_n)\exp\{- nP(X > b_n)\} = \infty$. The restrictions on the $b_n's$ can be removed.