  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 $x\to\infty$.
