On Exceedance Times for Some Processes with Dependent Increments

Abstract

Let {Z n } n≥0 be a random walk with a negative drift and i.i.d. increments with heavy-tailed distribution and let M = sup n≥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 → ∞, for the distribution of the quadruple that includes the time τ = τ (x) to exceed level x, position Z τ at this time, position Z τ −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 include 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