Journal of Applied Probability

On exceedance times for some processes with dependent increments

Søren Asmussen and Sergey Foss

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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).

Article information

Source
J. Appl. Probab., Volume 51, Number 1 (2014), 136-151.

Dates
First available in Project Euclid: 25 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1395771419

Digital Object Identifier
doi:10.1239/jap/1395771419

Mathematical Reviews number (MathSciNet)
MR3189447

Zentralblatt MATH identifier
1296.60120

Subjects
Primary: 60K15: Markov renewal processes, semi-Markov processes 60F10: Large deviations
Secondary: 60E99: None of the above, but in this section 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Björk-Grandell model Breiman's theorem conditioned limit theorem Markov modulation mean excess function random walk regenerative process regular variation ruin time subexponential distribution

Citation

Asmussen, Søren; Foss, Sergey. On exceedance times for some processes with dependent increments. J. Appl. Probab. 51 (2014), no. 1, 136--151. doi:10.1239/jap/1395771419. https://projecteuclid.org/euclid.jap/1395771419


Export citation