Subexponential asymptotics for stochastic processes : extremal behavior, stationary distributions and first passage probabilities
(1998) In Annals of Applied Probability 8(2). p.354374 Abstract
 Consider a reflected random walk Wn+1 = (Wn +Xn)(+), where Xo, 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 mu exceeds x is approximately mu (F) over bar(x) as x > infinity, and thereby that max (Wo,..., Wn) has the same asymptotics as max(Xo,...,Xn) 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 = (Wn +Xn)(+), where Xo, 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 mu exceeds x is approximately mu (F) over bar(x) as x > infinity, and thereby that max (Wo,..., Wn) has the same asymptotics as max(Xo,...,Xn) 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 infinitehorizon 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 lognormallike] that this leads to approximations for finitehorizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1273191
 author
 Asmussen, Sören ^{LU}
 organization
 publishing date
 1998
 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
 10505164
 language
 English
 LU publication?
 yes
 id
 95bf09618299435f9feeb453b80a55eb (old id 1273191)
 alternative location
 http://www.jstor.org/sici?sici=10505164(199805)8%3A2%3C354%3ASAFSPE%3E2.0.CO%3B2C&origin=ISI
 date added to LUP
 20081209 14:32:29
 date last changed
 20180529 12:03:01
@article{95bf09618299435f9feeb453b80a55eb, abstract = {Consider a reflected random walk Wn+1 = (Wn +Xn)(+), where Xo, 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 mu exceeds x is approximately mu (F) over bar(x) as x > infinity, and thereby that max (Wo,..., Wn) has the same asymptotics as max(Xo,...,Xn) 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 infinitehorizon 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 lognormallike] that this leads to approximations for finitehorizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized.}, author = {Asmussen, Sören}, issn = {10505164}, 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 = {354374}, 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}, }