On explicit form of the stationary distributions for a class of bounded\n Markov chains

2014/12/03 by Shaun McKinlay, McKinlay, Shaun, Konstantin Borovkov +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #45B05 #60J05 #60J20 #Bayesian Modeling and Causal Inference #FOS: Mathematics #Gene Regulatory Network Analysis #Markov Chains and Monte Carlo Methods #Petri Nets in System Modeling #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1412.1278

openalex publication_date 2014/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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