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On Exceedance Times for Some Processes with Dependent Increments

Part of:Distribution theory - ProbabilityLimit theoremsSpecial processes

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 {Z n }n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = supn≥0 Z n 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 Z τ at this time, position Z τ-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 processes60F10: Large deviations
Secondary:60E99: None of the above, but in this section60K25: Queueing theory

Information

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.

References

Albrecher, H., Asmussen, S. and Kortschak, D. (2012). Tail asymptotics for dependent subexponential differences. Siberian Math. J. 53, 965–983.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.Google Scholar
Asmussen, S. and Biard, R. (2011). Ruin probabilities for a regenerative Poisson gap generated risk process. Europ. Actuarial J. 1, 3–22.Google Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.Google Scholar
Asmussen, S. and Kortschak, D. (2011). Second order approximations for normalized ruin times in the presence of heavy tails. Stoch. Systems 3, 591–633.Google Scholar
Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at subexponential times, with queueing applications. Stoch. Process. Appl. 79, 265–286.Google Scholar
Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential Jumps. Adv. Appl. Prob. 31, 422–447.Google Scholar
Baccelli, F. and Foss, S. (2004). Moments and tails in monotone-separable stochastic networks. Ann. Appl. Prob. 14, 612–650.CrossRefGoogle Scholar
Balkema, A. A. and de Haan, L. (1974). Residual life time at great age. Ann. Prob. 2, 792–804.Google Scholar
Balkema, A. A., Resnick, S. I. and Klüppelberg, C. (1993). Densities with Gaussian tails. Proc. London Math. Soc. 66, 568–588.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Björk, T. and Grandell, J. (1988). Exponential inequalities for ruin probabilities in the Cox case. Scand. Actuarial J. 1988, 77–111.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323–331.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
Denisov, D. and Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046.Google Scholar
Denisov, D., Foss, S. and Korshunov, D. (2010). Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16, 971–994.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Springer, Berlin.Google Scholar
Foss, S. and Zachary, S. (2002). Asymptotics for the maximum of a modulated random walk with heavy-tailed increments. In Analytic Methods in Applied Probability (Amer. Math. Soc. Transl. Ser. 2 207), American Mathematical Society, Providence, RI, pp. 37–52.Google Scholar
Foss, S., Konstantopoulos, T. and Zachary, S. (2007). Discrete and continuous time modulated random walks with heavy-tailed increments. J. Theoret. Prob. 20, 581–612.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York.Google Scholar
Goldie, C. M. and Resnick, S. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution. Adv. Appl. Prob. 20, 706–718.Google Scholar
Klüppelberg, C., Kyprianou, A. and Maller, R. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar