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Maximum on a random time interval of a random walk with infinite mean
Denis Denisov · 2019-07-21 · via math.PR updates on arXiv.org

Let $ξ_1,ξ_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|ξ_1|]=\infty.$ Consider a random walk $S_n=ξ_1+\cdots+ξ_n$, a stopping time $τ=\min\{n\ge 1: S_n\le 0\}$ and let $M_τ=\max_{0\le i\le τ} S_i$. We study the asymptotics for $\mathbf P(M_τ>x),$ as $x\to\infty$.

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