Subexponential asymptotics for stochastic processes : extremal behavior, stationary distributions and first passage probabilities
Consider a reflected random walk Wn+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 --> inf