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

Consider a reflected random walk W(n+1) = (W(n) +X(n))(+), where X(o), X(1),... 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 (F) over bar(x) as x --> infinity, and thereby that max (W(o),..., W(n)) has the same asymptotics as max(X(o),...,X(n)) as n --> infinity. 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 Levy 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 he any value in [0, infinity]); 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.