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On Exceedance Times for Some Processes with Dependent Increments

By Søren Asmussen and Sergey Foss
Thiele Research Reports
No. 02, May 2012

Let $\{ Z_n\}_{n\ge 0}$ be a random walk with a negative drift and i.i.d. increments with heavy-tailed distribution and let $M=\sup\nolimits_{n\ge 0}Z_n$ be its supremum. Asmussen & Klüppelberg (1996) considered the behavior of the random walk given that $M > x$, for $x$ large, and obtained a limit theorem, as $x\to\infty$, for the distribution of the quadruple that includes the time $\tau=\tau(x)$ to exceed level $x$, position $Z_{\tau}$ at this time, position $Z_{\tau-1}$ 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 $\tau$. The class of models include Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of $\tau$ 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 & Schmidt (1999), Foss & Zachary (2002), and Foss, Konstantopoulos & Zachary (2007).

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

2010 Mathematics Subject Classification: Primary 60K15, 60F10, Secondary 60E99, 60K25

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