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Conditional limit theorems for spectrally positive Lévy processes

Part of: Stochastic processes Limit theorems Real functions

Published online by Cambridge University Press:  19 February 2016

Takis Konstantopoulos  and
Gregory S. Richardson
Takis Konstantopoulos
Affiliation:
University of Texas at Austin
Gregory S. Richardson*
Affiliation:
St Paul Companies
*
** Postal address: St Paul Companies, 310 N. Washington, 510C, St Paul, MN 55102, USA.
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Abstract

We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.

Keywords

Lévy processspectrally positiveinfinitely divisibleSkorokhod reflectionregular variationweak convergencelarge deviations

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60F17: Functional limit theorems; invariance principles
Secondary: 60G55: Point processes 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 158 - 178
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

*

Current address: Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712, USA. Email address: takis@ece.utexas.edu

This work was supported in part by NSF grant ANI-9903495 and was the basis of the second author's PhD dissertation (Richardson (2000)).

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