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Tail probabilities for non-standard risk and queueing processes with subexponential jumps

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen ,
Hanspeter Schmidli  and
Volker Schmidt
Søren Asmussen*
Affiliation:
University of Lund
Hanspeter Schmidli*
Affiliation:
Aarhus University
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden.
∗∗ Postal address: Institute of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: schmidli@imf.au.dk
∗∗∗ Postal address: Institute of Stochastics, University of Ulm, D-89069 Ulm, Germany.
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Abstract

A well-known result on the distribution tail of the maximum of a random walk with heavy-tailed increments is extended to more general stochastic processes. Results are given in different settings, involving, for example, stationary increments and regeneration. Several examples and counterexamples illustrate that the conditions of the theorems can easily be verified in practice and are in part necessary. The examples include superimposed renewal processes, Markovian arrival processes, semi-Markov input and Cox processes with piecewise constant intensities.

Keywords

Ruin probabilitystationary waiting time distributionrandom walkergodic inter-occurrence timesintegrated tail distributionregenerative surplus processregular variationsubexponential distribution

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 422 - 447
Copyright
Copyright © Applied Probability Trust 1999 

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