Maximum on a random time interval of a random walk with infinite mean

Abstract: Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min{n\ge 1: S_n\le 0}$ and let $M_\tau=\max_{0\le i\le \tau} S_i$. We study the asymptotics for $\mathbf P(M_\tau>x),$ as $x\to\infty$.