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March 2014 On exceedance times for some processes with dependent increments
Søren Asmussen, Sergey Foss
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J. Appl. Probab. 51(1): 136-151 (March 2014). DOI: 10.1239/jap/1395771419

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).

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Søren Asmussen. Sergey Foss. "On exceedance times for some processes with dependent increments." J. Appl. Probab. 51 (1) 136 - 151, March 2014. https://doi.org/10.1239/jap/1395771419

Information

Published: March 2014
First available in Project Euclid: 25 March 2014

zbMATH: 1296.60120
MathSciNet: MR3189447
Digital Object Identifier: 10.1239/jap/1395771419

Subjects:
Primary: 60F10, 60K15
Secondary: 60E99, 60K25

Rights: Copyright © 2014 Applied Probability Trust

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Vol.51 • No. 1 • March 2014
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