On a problem of Erdös and Taylor

Abstract

Let $\{S_n, n \geq 0\}$ be a centered d-dimensional random walk $(d \geq 3)$ and consider the so-called future infima process $J_n \stackrel{\mathrm{df}}{=}\inf _{k \geq n} \|S_k\|$. This paper is concerned with obtaining precise integral criteria for a function to be in the Lévy upper class of J. This solves an old problem of Erdös and Taylor, who posed the problem for the simple symmetric random walk on $\mathbb{Z}^d, d \geq 3$. These results are obtained by a careful analysis of the future infima of transient Bessel processes and using strong approximations. Our results belong to a class of Ciesielski-Taylor theorems which relate d-and $(d-2)$-dimensional Bessel processes.