On exceedance times for some processes with dependent increments

Soren Asmussen, Serguei Foss

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

Let {Z}>o be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = sup>0 Zn be its supremum. Asmussen and Kliippelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → oo, for the distribution of the quadruple that includes the time x - T(X) to exceed level x, position ZT at this time, position ZT i at the prior time, and the trajectory up to it (similar results were obtained for the Cramer-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 T. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Bjork-Grandell risk process, give examples where the order of x 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).

Original languageEnglish
Pages (from-to)136-151
Number of pages16
JournalJournal of Applied Probability
Volume51
Issue number1
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • Bjork-Grandell model
  • Breiman's theorem
  • Conditioned limit theorem
  • Markov modulation
  • Mean excess function
  • Random walk
  • Regenerative process
  • Regular variation
  • Ruin time
  • Subexponential distribution

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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