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

Sören Asmussen

Research output: Contribution to journalArticlepeer-review

120 Citations (SciVal)

Abstract

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 --> 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.
Original languageEnglish
Pages (from-to)354-374
JournalAnnals of Applied Probability
Volume8
Issue number2
Publication statusPublished - 1998

Subject classification (UKÄ)

  • Probability Theory and Statistics

Keywords

  • cycle maximum
  • extremal index
  • extreme values
  • Frechet distribution
  • Gumbel distribution
  • interest force
  • level crossings
  • maximum domain of attraction
  • overshoot distribution
  • random walk
  • rare event
  • regular variation
  • ruin probability
  • stable process
  • storage process
  • subexponential distribution
  • RUIN
  • QUEUE

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