## Journal of Applied Probability

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

### On exceedance times for some processes with dependent increments

Søren Asmussen and Sergey Foss

#### Abstract

Let {*Z*_{n}}_{n≥0} be a random walk with a negative drift and
independent and identically distributed increments with heavy-tailed distribution, and let
*M* = sup_{n≥0}*Z*_{n} 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
*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 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

**Source**

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

**Dates**

First available in Project Euclid: 25 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1395771419

**Digital Object Identifier**

doi:10.1239/jap/1395771419

**Mathematical Reviews number (MathSciNet)**

MR3189447

**Zentralblatt MATH identifier**

1296.60120

**Subjects**

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]

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.jap/1395771419