On exceedance times for some processes with dependent increments

TL;DR

This paper extends existing limit theorems for the exceedance times of heavy-tailed random walks with negative drift to models with dependent increments, including Markov-modulated and fluid models, providing a broader understanding of their asymptotic behavior.

Contribution

It introduces new limit theorems for exceedance times in dependent increment models, expanding the scope beyond independent cases and including various regenerative and Markov-modulated processes.

Findings

Limit theorems for exceedance times in dependent models

Analysis of growth rates of exceedance times

Extensions to fluid and risk process models

Abstract

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 $M=\sup_{n\ge 0}Z_n$ be its supremum. Asmussen & Kl{\"u}ppelberg (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 $\rtreg=\rtreg(x)$ to exceed level $x$, position $Z_{\rtreg}$ at this time, position $Z_{\rtreg-1}$ at the prior time, and the trajectory up to it (similar results were obtained for the Cram\'er-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 Bj{\"o}rk-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).