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Large Deviations for Point Processes Based on Stationary Sequences with Heavy Tails

Part of: Limit theorems Probability theory on algebraic and topological structures Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Henrik Hult  and
Gennady Samorodnitsky
Henrik Hult*
Affiliation:
KTH
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
∗Postal address: Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden. Email address: [email protected]
∗∗Postal address: School of Operations Research and Industrial Engineering, Cornell University, 220 Rhodes Hall, Ithaca, NY 14853, USA. Email address: [email protected]
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Abstract

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In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.

Keywords

Stationary sequenceregular variationlarge deviationspoint process

MSC classification

Secondary: 60F10: Large deviations 60G10: Stationary processes 60G55: Point processes 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 1 - 40
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research partially supported by the Swedish Research Council.

Research partially supported by NSA grant H98230-06-1-0069 and ARO grant W911NF-07-1-0078 at Cornell University.

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