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

Let ${\xi }_1,{\xi }_2,\dots$ be independent, identically distributed random variables with infinite mean $\mathbf{E}[\mathrm{\mid }{\xi }_1\mathrm{\mid }]=\mathrm{\infty }\mathrm{.}$ 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 $x\to \mathrm{\infty }$.