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On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

Part of: Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Remigijus Leipus ,
Vygantas Paulauskas  and
Donatas Surgailis
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Vygantas Paulauskas*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
*
∗Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.
∗Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.
∗∗∗∗Postal address: Institute of Mathematics and Informatics, Akademijos str. 4, LT-08663 Vilnius, Lithuania. Email address: [email protected]
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Abstract

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We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.

Keywords

AR(1) modelregime switchingrenewal-reward processstable Lévy process

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 421 - 440
Copyright
© Applied Probability Trust 2006 

Footnotes

Supported by bilateral Lithuania-France research project Gilibert and the Lithuanian State Science and Studies Foundation grant T-10/06.

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