Local Asymptotics of the Cycle Maximum of a Heavy-Tailed Random Walk

Let ??, ?????, ?????,... be a sequence of independent and identically distributed random variables, and let \$S\_\{n\}={\textbackslash}xi \_\{1\}+{\textbackslash}cdots +{\textbackslash}xi \_\{n\}\$ and \$M\_\{n\}={\textbackslash}text\{max\}\_\{k{\textbackslash}leq n\}S\_\{k\}\$. Let \${\textbackslash}tau ={\textbackslash}text\{min\}{\textbackslash}\{n{\textbackslash}geq 1{\textbackslash}colon S\_\{n\}{\textbackslash}leq 0{\textbackslash}\}\$. We assume that ?? has a heavy-tailed distribution and negative, finite mean E(??) \< 0. We find the asymptotics of \${\textbackslash}text\{P\}{\textbackslash}\{M\_\{{\textbackslash}tau\}{\textbackslash}in (x,x+T]{\textbackslash}\}\$ as x ??? ???, for a fixed, positive constant T.