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On Exceedance Times for Some Processes with Dependent Increments

Part of: Special processes Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  30 January 2018

Søren Asmussen  and
Sergey Foss
Søren Asmussen*
Affiliation:
Aarhus University
Sergey Foss*
Affiliation:
Heriot-Watt University and Sobolev Institute of Mathematics
*
∗ Postal address: Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: asmus@imf.au.dk
∗∗ Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@hw.ac.uk
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Abstract

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Let {Zn}n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = supn≥0Zn be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Zτ at this time, position Zτ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

Keywords

Björk-Grandell modelBreiman's theoremconditioned limit theoremMarkov modulationmean excess functionrandom walkregenerative processregular variationruin timesubexponential distribution

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60F10: Large deviations
Secondary: 60E99: None of the above, but in this section 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 136 - 151
Copyright
© Applied Probability Trust 

Footnotes

This author thanks the Kazakhstan Ministry of Education and Science (grant 1778/GF) for partial support.

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