Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T03:37:56.776Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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)).

References

Anantharam, V. (1988). How large delays build up in a GI/G/1 queue. Queueing Systems 5, 345–368.Google Scholar
Asmussen, S. (1996). Rare events in the presence of heavy tails. In Stochastic Networks: Rare Events and Stability (Lecture Notes Statist. 117), eds Glasserman, P., Sigman, K. and Yao, D. D., Springer, New York, pp. 197–214.Google Scholar
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
Chen, H. and Whitt, W. (1993). Diffusion approximations for open queueing networks with service interruptions. Queueing Systems 13, 335–359.Google Scholar
Durrett, R. (1980). Conditioned limit theorems for random walks with negative drift. Z. Wahrscheinlichkeitsth. 52, 277–287.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. (1981). Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure. Ann. Prob. 9, 468–481.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Harrison, J. M. (1977). The supremum distribution of a Lévy process with no negative jumps. Adv. Appl. Prob. 9, 417–422.CrossRefGoogle Scholar
Konstantopoulos, T. (1999). The Skorokhod reflection problem for functions with discontinuities. Tech. Rep., Department of Electrical and Computer Engineering, University of Texas at Austin.Google Scholar
Konstantopoulos, T. and Last, G. (2000). On stationary reflected Lévy processes. Preprint, ECE Department, University of Texas at Austin.Google Scholar
Konstantopoulos, T. and Lin, S. J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28, 215–243.CrossRefGoogle Scholar
Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Preprint. To appear in Ann. Appl. Prob.Google Scholar
Norros, I. (1999). Busy periods of fractional Brownian storage: a large deviations approach. Adv. Performance Anal. 2, 1–19.Google Scholar
O'Brien, G. (1999). Unusually large values for spectrally positive stable and related processes. Ann. Prob. 27, 990–1008.Google Scholar
Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer, Berlin.Google Scholar
Richardson, G. S. (2000). Rare events and conditional limit theorems for a class of spectrally positive, heavy-tailed Lévy processes. , University of Texas at Austin.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies Adv. Math. 68). Cambridge University Press.Google Scholar
Takács, L., (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar
Whitt, W. (1999). The reflection map is Lipschitz with appropriate Skorohod M metrics. Tech. Rep., AT&T Labs.Google Scholar
Zolotarev, V. M. (1964). The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–662.Google Scholar