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A new theorem on the existence of invariant distributions with applications to ARCH processes

Part of: Applications Markov processes

Published online by Cambridge University Press:  14 July 2016

Vytautas Kazakevičius  and
Remigijus Leipus
Vytautas Kazakevičius*
Affiliation:
Vilnius University
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics
*
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania.
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania.
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Abstract

A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.

Keywords

Markov chaininvariant distributionintegrated ARCH processnonlinear time series

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 147 - 162
Copyright
Copyright © Applied Probability Trust 2003 

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