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Random coefficient autoregression, regime switching and long memory

Part of: Special processes Limit theorems Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Remigijus Leipus  and
Donatas Surgailis
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Email address: [email protected]
∗∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania.
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Abstract

We discuss long-memory properties and the partial sums process of the AR(1) process {Xt, t ∈ 𝕫} with random coefficient {at, t ∈ 𝕫} taking independent values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic Aj has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {Xt} decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {Xt} weakly converges to a stable Lévy process.

Keywords

Random coefficient autoregressionregime switchingrenewal processstable Lévy motionlong memory

MSC classification

Primary: 60K05: Renewal theory 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60F05: Central limit and other weak theorems 60G52: Stable processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 737 - 754
Copyright
Copyright © Applied Probability Trust 2003 

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