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

TL;DR

This paper derives explicit stationary distributions for a class of bounded Markov chains on [0,1], with applications including robot coverage algorithms, under broad ergodicity conditions and specific beta-distributed jump proportions.

Contribution

It provides closed-form expressions for stationary densities of these Markov chains when jump proportions follow a beta distribution with first parameter 1, extending previous studies.

Findings

Established ergodicity under broad conditions

Derived explicit stationary density formulas

Demonstrated applications in robot coverage

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 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.