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May 1998 Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities
Søren Asmussen
Ann. Appl. Probab. 8(2): 354-374 (May 1998). DOI: 10.1214/aoap/1028903531

Abstract

Consider a reflected random walk Wn+1=(Wn+Xn)+, where X0,X1, 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 μ exceeds x is approximately μF¯(x) as x, and thereby that max(W0,,Wn) has the same asymptotics as max(X0,,Xn) as n. In particular, the extremal index is shown to be θ=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,]; 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.

Citation

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Søren Asmussen. "Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities." Ann. Appl. Probab. 8 (2) 354 - 374, May 1998. https://doi.org/10.1214/aoap/1028903531

Information

Published: May 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0942.60034
MathSciNet: MR1624933
Digital Object Identifier: 10.1214/aoap/1028903531

Subjects:
Primary: 60G70 , 60K25 , 60K30

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

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.8 • No. 2 • May 1998
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