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Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

F. Simonot  and
Y. Q. Song
F. Simonot*
Affiliation:
Université Henri Poincaré
Y. Q. Song*
Affiliation:
CRIN-ENSEM
*
∗Postal address: ESSTIN-Université Henri Poincaré, Nancy 1, Parc R Bentz, 54 500 Vandoeuvre, France. email:francois@ensem.u-nancy.fr
∗∗Postal address: CRIN-ENSEM, 2, av. de la Forêt de Haye, 54 516 Vandoeuvre, France. email:song@ensem.u-nancy.fr
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Abstract

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

Keywords

MARKOV CHAINSTOCHASTIC MONOTONICITYRIGHT-BANDED MATRIXGEOMETRIC RECURRENCECONVERGENCE RATERANDOM WALKLINDLEY PROCESS

MSC classification

Secondary: 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 974 - 985
Copyright
Copyright © Applied Probability Trust 1996 

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