The Growth of Random Walks and Levy Processes

Abstract

Let $\{X_i\}$ be a sequence of independent, identically distributed non-degenerate random variables taking values in $\mathbb{R}^d$ and $S_n = \sum^n_{i = 1} X_i, M_n = \max_{1\leqq i \leqq n} |S_i|$. Define for $x > 0, G(x) = P\{| X_1 | > x\}, K(x) = x^{-2}E(| X_1 |^2 1\{| X_1 | \leq x\}), M(x) = x^{-1} |E(X_1 1\{| X_1 | \leq x\})|,$ and $h(x) = G(x) + K(x) + M(x)$. Then if $\beta = \sup \{\alpha: \lim \sup x^\alpha h(x) = 0\}, \delta = \sup \{\alpha: \lim \inf x^\alpha h(x) = 0\}$, it is proved that $n^{-1/\alpha}M_n \rightarrow 0$ for $\alpha < \beta, \rightarrow \infty$ for $\alpha > \delta$, while the $\lim \inf$ is 0 and the $\lim \sup$ is $\infty$ for $\beta < \alpha < \delta$. Some alternative characterizations of the indices $\beta, \delta$ are obtained as well as the analogous results for Levy processes.