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

Abstract

AbstractLet $$\xi _1,\xi _2,\ldots $$ ξ 1 , ξ 2 , … be independent, identically distributed random variables with infinite mean $${\mathbf {E}}[|\xi _1|]=\infty .$$ E [ | ξ 1 | ] = ∞ . Consider a random walk $$S_n=\xi _1+\cdots +\xi _n$$ S n = ξ 1 + ⋯ + ξ n , a stopping time $$\tau =\min \{n\ge 1: S_n\le 0\}$$ τ = min { n ≥ 1 : S n ≤ 0 } and let $$M_\tau =\max _{0\le i\le \tau } S_i$$ M τ = max 0 ≤ i ≤ τ S i . We study the asymptotics for $${\mathbf {P}}(M_\tau >x),$$ P ( M τ > x ) , as $$x\rightarrow \infty $$ x → ∞ .