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Fractional Moments of Solutions to Stochastic Recurrence Equations

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Thomas Mikosch ,
Gennady Samorodnitsky  and
Laleh Tafakori
Thomas Mikosch*
Affiliation:
University of Copenhagen
Gennady Samorodnitsky*
Affiliation:
Cornell University
Laleh Tafakori*
Affiliation:
Shiraz University
*
∗ Postal address: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. 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]
∗∗∗ Postal address: Department of Statistics, Shiraz University, College of Sciences, Shiraz, 7146713565, Iran. Email address: [email protected]
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Abstract

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In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

Keywords

Momentstochastic recurrence equationGARCHErlang distributionnumerical approximation

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 969 - 982
Copyright
© Applied Probability Trust 

Footnotes

Research partly supported by the Danish Natural Science Research Council (FNU) grant 10-084172 ‘Heavy tail phenomena: Modeling and estimation’.

Research partially supported by the ARO grant W911NF-07-1-0078, NSF grant DMS-1005903, and NSA grant H98230-11-1-0154 at Cornell University.

This paper was written when Laleh Tafakori visited the Department of Mathematics at the University of Copenhagen in 2011. She takes pleasure to thank the host institution for its hospitality.

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