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Subgeometric rates of convergence for Markov processes under subordination

Published online by Cambridge University Press:  17 March 2017

Chang-Song Deng*
Affiliation:
Wuhan University
René L. Schilling*
Affiliation:
TU Dresden
Yan-Hong Song*
Affiliation:
Zhongnan University of Economics and Law
*
* Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email address: [email protected]
** Postal address: Fachrichtung Mathematik, Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany. Email address: [email protected]
*** Postal address: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China. Email address: [email protected]

Abstract

We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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