Asymptotics for the maximum of a modulated random walk with heavy-tailed increments

Abstract

We consider asymptotics for the maximum of a modulated random walk whose increments ${\xi }_n^{X_n}$ are heavy-tailed. Of particular interest is the case where the modulating process $X$ is regenerative. Here we study also the maximum of the recursion given by $W_0=0$ and, for $n\ge 1$, $W_n=\max (0,W_{n-1}+{\xi }_n^{X_n})$

19 pages, In Analytic Methods in Applied Probability: In Memory of Fridrih Karpelevich (edited by Yu. M. Suhov), American Mathematical Society (2002)