A Note on the Asymptotics for the Maximum on a Random Time Interval of a Random Walk

#### D. Denisov

2005, v.11, Issue 1, 165-169

ABSTRACT

Let $\xi,\xi_1$, $\xi_2$, \ldots\ be independent random variables with a common distribution $F$ and negative mean $\E\xi=-m$. Consider the random walk $S_0=0$, $S_n=\xi_1+\cdots+\xi_n$, the stopping time $\tau = \min\{n\ge 1: S_n\le 0\}$ and let $M_\tau=\max_{0\le i\le \tau} S_i.$ S. Asmussen found the asymptotics of $\P(M_\tau>x)$ as $x \to\infty$, when $F$ is subexponential. We give a short proof of this result. We also provide comments on the light-tailed case.

Keywords: random walk,cycle maximum,heavy-taileddistribution,stopping time