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

Asmussen, Sören LU (1998) In Annals of Applied Probability 8(2). p.354-374
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
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
in
Annals of Applied Probability
volume
8
issue
2
pages
354 - 374
publisher
Institute of Mathematical Statistics
external identifiers
  • scopus:0032221191
ISSN
1050-5164
language
English
LU publication?
yes
id
95bf0961-8299-435f-9fee-b453b80a55eb (old id 1273191)
alternative location
http://www.jstor.org/sici?sici=1050-5164(199805)8%3A2%3C354%3ASAFSPE%3E2.0.CO%3B2-C&origin=ISI
date added to LUP
2008-12-09 14:32:29
date last changed
2017-03-19 04:21:19
@article{95bf0961-8299-435f-9fee-b453b80a55eb,
  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.},
  author       = {Asmussen, Sören},
  issn         = {1050-5164},
  keyword      = {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},
  language     = {eng},
  number       = {2},
  pages        = {354--374},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Applied Probability},
  title        = {Subexponential asymptotics for stochastic processes : extremal behavior, stationary distributions and first passage probabilities},
  volume       = {8},
  year         = {1998},
}