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

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.

On a class of random walks in simplexes

“…In [3], Diaconis and Freedman reconsidered Sethuraman's model from the point of view of random iterated functions, and also studied the case where p(z) depends on z ∈ S 1 = [0, 1]. Other models in S 1 with various special cases of p(z) and ξ were studied in [7], [8], and [9]. Inspired by the work of Diaconis and Freedman, Ladjimi and Peigné in their recent work [6] studied iterated random functions with place-dependent probability choice functions, and demonstrated several applications to the one-dimensional model where ξ ∼ Uniform[0, 1], and p(z) is a Hölder-continuous function in [0,1].…”

“…In [7], McKinlay and Borovkov gave a general condition for the ergodicity of the onedimensional Markov chain {Z n } n≥0 in S 1 . By solving integral equations, they derived a closedform expression for the stationary density function in the case where ξ ∼ Beta(1, γ ), and p(z) is a piecewise continuous function on [0, 1].…”

“…The model, also known in the literature as a stick-breaking process, a stochastic give-andtake (see [2], [7]) or a Diaconis-Freedman chain (see [6]) has many applications in other fields such as human genetics, robot coverage algorithms, random search, etc. For further discussions, we refer the reader to [2], [9], and [7].…”

On beta distributed limits of iterated linear random functions

“…Therefore, it suffices to show that X ∼ β((1 − p)z, pz) is the unique solution to (26) with (C, D) distributed as per (28). Uniqueness follows from Lemma 1.5 in [40] since condition (7) is satisfied for an i.i.d.…”

“…[8,10,37,38]). See also [32,26], where an extension of this model to the case when the direction of the next move is a function of the particles current location was considered.…”