Journal of Applied Probability

On exceedance times for some processes with dependent increments

Søren Asmussen and Sergey Foss

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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).

Article information

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

First available in Project Euclid: 25 March 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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