Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities

Abstract

Consider a reflected random walk $W_{n+1} = (W_n + X_n)^+$, where $X_0, X_1,\dots$ are i.i.d. with negative mean and subexponential with common distribution F. It is shown that the probability that the maximum within a regenerative cycle with mean $\mu$ exceeds x is approximately $\mu\bar{F}(x)$ as $x \to \infty$, and thereby that $\max(W_0, \dots, W_n)$ has the same asymptotics as $\max(X_0, \dots, X_n)$ as $n \to \infty$. In particular, the extremal index is shown to be $\theta = 0$, and the point process of exceedances of a large level is studied. The analysis extends to reflected Lévy processes in continuous time, say, stable processes. Similar results are obtained for a storage process with release rate $r(x)$ at level x and subexponential jumps (here the extremal index may be any value in $[0, \infty]$; also the tail of the stationary distribution is found. For a risk process with premium rate $r(x)$ at level x and subexponential claims, the asymptotic form of the infinite-horizon ruin probability is determined. It is also shown by example $[r(x) = a + bx$ and claims with a tail which is either regularly varying, Weibull- or lognormal-like] that this leads to approximations for finite-horizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized.