March 2016 On explicit form of the stationary distributions for a class of bounded Markov chains
S. McKinlay, K. Borovkov
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J. Appl. Probab. 53(1): 231-243 (March 2016).

Abstract

We consider a class of discrete-time Markov chains with state space [0, 1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were the subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed-form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

Citation

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S. McKinlay. K. Borovkov. "On explicit form of the stationary distributions for a class of bounded Markov chains." J. Appl. Probab. 53 (1) 231 - 243, March 2016.

Information

Published: March 2016
First available in Project Euclid: 8 March 2016

zbMATH: 1337.60167
MathSciNet: MR3471959

Subjects:
Primary: 60J05
Secondary: 45B05 , 60J20

Keywords: Beta distribution , ergodicity , give-and-take model , Markov chain , random search , semidegenerate kernel , stationary distribution

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 1 • March 2016
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