The Annals of Applied Probability

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

Søren Asmussen

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Abstract

Consider a reflected random walk $W_{n+1} = (W_n + X_n)^+$, where $X_0, X_1,\dots$ 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 $\mu\bar{F}(x)$ as $x \to \infty$, and thereby that $\max(W_0, \dots, W_n)$ has the same asymptotics as $\max(X_0, \dots, X_n)$ as $n \to \infty$. 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 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 $[0, \infty]$; 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.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 2 (1998), 354-374.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903531

Digital Object Identifier
doi:10.1214/aoap/1028903531

Mathematical Reviews number (MathSciNet)
MR1624933

Zentralblatt MATH identifier
0942.60034

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60K25: Queueing theory [See also 68M20, 90B22]

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

Citation

Asmussen, Søren. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 (1998), no. 2, 354--374. doi:10.1214/aoap/1028903531. https://projecteuclid.org/euclid.aoap/1028903531


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