Asymptotic distributions for the Ehrenfest urn and related random walks

The distributions of nearest neighbour random walks on hypercubesin continuous timet0 can be expressed in terms of binomial distributions; their limit behaviour fort, N →∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity forN →∞.Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.

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Composition Markov chains of multinomial type

Advances in Applied Probability ◽

10.1017/s0001867800003220 ◽

2009 ◽

Vol 41 (01) ◽

pp. 270-291 ◽

Cited By ~ 2

Author(s):

Hua Zhou ◽

Kenneth Lange

Keyword(s):

Markov Chain ◽

Random Walks ◽

Continuous Time ◽

Spectral Decomposition ◽

Equilibrium Distribution ◽

Convergence To Equilibrium ◽

Light Bulb ◽

Krawtchouk Polynomials ◽

Particle Counts ◽

Coalescence Times

Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

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AN IDENTIFICATION PROBLEM IN AN URN AND BALL MODEL WITH HEAVY TAILED DISTRIBUTIONS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990143 ◽

2009 ◽

Vol 24 (1) ◽

pp. 77-97

Author(s):

Christine Fricker ◽

Fabrice Guillemin ◽

Philippe Robert

Keyword(s):

Basic Problem ◽

Identification Problem ◽

Total Variation Norm ◽

Heavy Tailed Distributions ◽

Ball Problem ◽

Original Number ◽

Explicit Bounds ◽

Variation Norm ◽

Heavy Tailed ◽

Stein Method

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.

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Asymptotic distributions of continuous-time random walks: A probabilistic approach

Journal of Statistical Physics ◽

10.1007/bf02179257 ◽

1995 ◽

Vol 81 (3-4) ◽

pp. 777-792 ◽

Cited By ~ 96

Author(s):

Marcin Kotulski

Keyword(s):

Random Walks ◽

Continuous Time ◽

Probabilistic Approach ◽

Asymptotic Distributions ◽

Continuous Time Random Walks

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Subgeometric rates of convergence for a class of continuous-time Markov process

Journal of Applied Probability ◽

10.1017/s0021900200000711 ◽

2005 ◽

Vol 42 (03) ◽

pp. 698-712

Author(s):

Zhenting Hou ◽

Yuanyuan Liu ◽

Hanjun Zhang

Keyword(s):

Markov Process ◽

Continuous Time ◽

Invariant Probability Measure ◽

Sufficient Conditions ◽

Rates Of Convergence ◽

Transition Function ◽

Total Variation Norm ◽

Rate Functions ◽

General State Space ◽

Explicit Bounds

Let (Φ t ) t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function P t (x, ·) to π; specifically, we find conditions under which r(t)||P t (x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

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Composition Markov chains of multinomial type

Advances in Applied Probability ◽

10.1239/aap/1240319585 ◽

2009 ◽

Vol 41 (1) ◽

pp. 270-291 ◽

Cited By ~ 11

Author(s):

Hua Zhou ◽

Kenneth Lange

Keyword(s):

Markov Chain ◽

Random Walks ◽

Continuous Time ◽

Spectral Decomposition ◽

Equilibrium Distribution ◽

Convergence To Equilibrium ◽

Light Bulb ◽

Krawtchouk Polynomials ◽

Particle Counts ◽

Coalescence Times

Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

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Subgeometric rates of convergence for a class of continuous-time Markov process

Journal of Applied Probability ◽

10.1239/jap/1127322021 ◽

2005 ◽

Vol 42 (3) ◽

pp. 698-712 ◽

Cited By ~ 10

Author(s):

Zhenting Hou ◽

Yuanyuan Liu ◽

Hanjun Zhang

Keyword(s):

Markov Process ◽

Continuous Time ◽

Invariant Probability Measure ◽

Sufficient Conditions ◽

Rates Of Convergence ◽

Transition Function ◽

Total Variation Norm ◽

Rate Functions ◽

General State Space ◽

Explicit Bounds

Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

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Numerical Investigation of Superslow Diffusion Laws for a Certain Class of Continuous-time Random Walks

Journal of Nano- and Electronic Physics ◽

10.21272/jnep.8(1).01044 ◽

2016 ◽

Vol 8 (1) ◽

pp. 01044-1-01044-5

Author(s):

Yu. S. Bystrik ◽

Keyword(s):

Random Walks ◽

Numerical Investigation ◽

Continuous Time ◽

Continuous Time Random Walks

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Strong Local Survival of Branching Random Walks is Not Monotone

Advances in Applied Probability ◽

10.1017/s000186780000714x ◽

2014 ◽

Vol 46 (02) ◽

pp. 400-421 ◽

Cited By ~ 1

Author(s):

Daniela Bertacchi ◽

Fabio Zucca

Keyword(s):

Random Walk ◽

Random Walks ◽

Generating Function ◽

Continuous Time ◽

Local Property ◽

Branching Random Walk ◽

Infinite Dimensional ◽

Branching Random Walks ◽

Local Survival ◽

Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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