The stationary probability density of a class of bounded Markov processes

Ramli, Muhamad Azfar; Leng, Gerard

doi:10.1239/aap/1293113147

Cited by 8 publications

(17 citation statements)

References 11 publications

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“…See for instance [Ramli and Leng, 2010] or [Ladjimi and Peigné, 2019]. We do not get such general result when d ≥ 2, we can do it only in some specific cases.…”

Section: Some Explicit Invariant Probability Densitiesmentioning

confidence: 86%

On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

Peigné,

Tran

2020

Preprint

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In this paper, we give out a setting of an Diaconis and Freedman's chain in a multidimensional simplex and consider its asymptotic behavior. By using techniques in random iterated functions theory and quasi-compact operators theory, we first give out some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure. In some particular cases, we give out explicit formulas of the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimensional two. Eventually, some other settings of the chain are discussed.

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“…See for instance [Ramli and Leng, 2010] or [Ladjimi and Peigné, 2019]. We do not get such general result when d ≥ 2, we can do it only in some specific cases.…”

Section: Some Explicit Invariant Probability Densitiesmentioning

confidence: 86%

On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

Peigné,

Tran

2020

Preprint

View full text Add to dashboard Cite

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“…This paper is mostly devoted to deriving explicit formulae for the stationary densities for a class of ergodic [0, 1]-valued discrete time Markov chains that appear in some interesting applications (see e.g. Section 4 in [7], Section 5 in [12], and Section 3 below). The chain dynamics are as follows.…”

Section: Introductionmentioning

confidence: 99%

“…The case of non-constant p(x) was also discussed, but not pursued in [5]. Further special cases of that model for various choices of p(x) and distributions F L , F R were considered in [16], [1], [15], [11] and [12]. In particular, [11] dealt with the case when p(x) ≡ x and F L = F R = β(1, z), z > 0, where β(a, b) denotes the beta distribution with density…”

Section: Introductionmentioning

confidence: 99%

“…It was shown in [11] that in that case X was ergodic with stationary law β(z, z). The case when F L = F R = U [0, 1] and p(x) is piecewise continuous was studied in [12].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, by choosing large enough l and r, one can obtain unimodal and multimodal stationary densities with arbitrarily high and "sharp" peaks. We will use this feature to generalise the robot coverage algorithm from [12] (see Section 3.3 below).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

McKinlay

Borovkov

2016

J. Appl. Probab.

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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.

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On beta distributed limits of iterated linear random functions

McKinlay

2017

Statistics & Probability Letters

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We consider several special cases of iterations of random i.i.d. linear functions with beta distributed fixed points that generate nested interval schemes when iterated in a backward direction, and ergodic Markov chains in the forward direction. We prove that the fixed points are beta distributed by using a related random equation with a gamma distributed solution that also generates a corresponding ergodic Markov chain with a gamma distributed stationary distribution. Our approach allows us to find limiting distributions of the random processes we consider, and provide solutions to the two random equations, by solving only one of these equations. The paper extends many partial results available in the existing literature.

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The stationary probability density of a class of bounded Markov processes

Cited by 8 publications

References 11 publications

On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

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

On beta distributed limits of iterated linear random functions

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