The Minimal Growth Rate of Partial Maxima

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables and let $M_n = \max_{i \leq j \leq n} X_j$. For each real sequence $\{b_n\}$, a sequence $\{b^\ast_n\}$ and a sub-sequence of integers $\{n_k\}$ is explicitly constructed such that $P(M_n \leq b_n \text{i.o.}) = 1 \operatorname{iff} \sum_k P(M_{n_k} \leq b^\ast_{n_k}) = \infty$. This result gives a complete characterization of the upper and lower-class sequences (as introduced by Paul Levy) for the a.s. minimal growth rate of $\{M_n\}$.