
Mathematics 2006
Tail asymptotics for the maximum of perturbed random walkDOI: 10.1214/105051606000000268 Abstract: Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a stationary sequence $(\xi_n:n\geq 0)$ to produce the process $(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the distribution of the alltime maximum $M_{\infty}=\max \{S_k+\xi_k:k\geq0\}$ of perturbed random walk with a negative drift. Such a maximum arises in several different applications settings, including production systems, communications networks and insurance risk. Our main results describe asymptotics for $\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics depend greatly on whether the $\xi_n$'s are lighttailed or heavytailed. In the lighttailed setting, the tail asymptotic is closely related to the Cram\'{e}rLundberg asymptotic for standard random walk.
