On the age distribution of a Markov chain

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

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Conditional limit theorems for a left-continuous random walk

Journal of Applied Probability ◽

10.2307/3212494 ◽

1973 ◽

Vol 10 (1) ◽

pp. 39-53 ◽

Cited By ~ 9

Author(s):

A. G. Pakes

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Absorbing State ◽

Conditional Limit Theorems ◽

Recurrent Markov Chain ◽

Continuous Random Walk ◽

Condition C ◽

Positive Recurrent ◽

Positive Integers ◽

Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

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Conditional limit theorems for a left-continuous random walk

Journal of Applied Probability ◽

10.1017/s0021900200042078 ◽

1973 ◽

Vol 10 (01) ◽

pp. 39-53 ◽

Cited By ~ 6

Author(s):

A. G. Pakes

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Absorbing State ◽

Conditional Limit Theorems ◽

Recurrent Markov Chain ◽

Continuous Random Walk ◽

Condition C ◽

Positive Recurrent ◽

Positive Integers ◽

Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

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Limit theorems for some continuous-time random walks

Advances in Applied Probability ◽

10.1239/aap/1316792670 ◽

2011 ◽

Vol 43 (3) ◽

pp. 782-813 ◽

Cited By ~ 1

Author(s):

M. Jara ◽

T. Komorowski

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Quantum Transport ◽

Continuous Time ◽

Limit Theorems ◽

Transport Theory ◽

Sufficient Conditions ◽

Continuous Time Random Walks ◽

Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

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Antiduality and Möbius monotonicity: generalized coupon collector problem

ESAIM Probability and Statistics ◽

10.1051/ps/2019004 ◽

2019 ◽

Vol 23 ◽

pp. 739-769

Author(s):

Paweł Lorek

Keyword(s):

Markov Chain ◽

State Space ◽

Partial Ordering ◽

Absorption Time ◽

Absorbing Markov Chain ◽

Coupon Collector ◽

Strong Stationary Duality ◽

Finite State ◽

A Chain ◽

Finite State Space

For a given absorbing Markov chain X* on a finite state space, a chain X is a sharp antidual of X* if the fastest strong stationary time (FSST) of X is equal, in distribution, to the absorption time of X*. In this paper, we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence – utilizing known results on the limiting distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its FSST is distributed as a prescribed mixture of sums of geometric random variables.

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On the maximum and absorption time of left-continuous random walk

Journal of Applied Probability ◽

10.2307/3213402 ◽

1978 ◽

Vol 15 (2) ◽

pp. 292-299 ◽

Cited By ~ 11

Author(s):

Anthony G. Pakes

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Initial Position ◽

Absorption Time ◽

Conditional Limit Theorems ◽

Negative Drift ◽

Continuous Random Walk ◽

Conditional Limit ◽

Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

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Some limit theorems for positive recurrent branching Markov chains: II

Advances in Applied Probability ◽

10.1017/s0001867800008569 ◽

1998 ◽

Vol 30 (03) ◽

pp. 711-722 ◽

Cited By ~ 3

Author(s):

Krishna B. Athreya ◽

Hye-Jeong Kang

Keyword(s):

Markov Chain ◽

Limit Theorems ◽

Law Of Large Numbers ◽

General State ◽

Position Distribution ◽

Large Numbers ◽

Recurrent Markov Chain ◽

General State Space ◽

Watson Process ◽

Positive Recurrent

In this paper we consider a Galton-Watson process in which particles move according to a positive recurrent Markov chain on a general state space. We prove a law of large numbers for the empirical position distribution and also discuss the rate of this convergence.

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On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

Journal of Applied Probability ◽

10.1017/s002190020010659x ◽

1992 ◽

Vol 29 (01) ◽

pp. 21-36 ◽

Cited By ~ 3

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chain ◽

Random Walk ◽

Markov Chains ◽

State Space ◽

Stationary Distribution ◽

Practical Importance ◽

Prime Concern ◽

Continuous Random Walk ◽

Transient Markov Chain ◽

Quasi Stationary Distribution

Let {Xn, n= 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩+= {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix𝐏= (pij), and letpij(n) =P∈Xn=j, Xk∈ 𝒩+fork= 0, 1, ···,n|X0=i],i, j𝒩+. The prime concern of this paper is conditions for the existence of the limits,qijsay, ofasn →∞. Ifthe distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vectorx= (xi) satisfyingrxT=xT𝐏andexists, whereris the convergence norm of𝐏, i.e.r=R–1andand T denotes transpose, then it is unique, positive elementwise, andqij(n) necessarily converge toxjasn →∞.Unlike existing results in the literature, our results can be applied even to theR-null andR-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix isR-transient is discussed to demonstrate the usefulness of our results.

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On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

Journal of Applied Probability ◽

10.2307/3214788 ◽

1992 ◽

Vol 29 (1) ◽

pp. 21-36 ◽

Cited By ~ 12

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chain ◽

Random Walk ◽

Markov Chains ◽

State Space ◽

Stationary Distribution ◽

Practical Importance ◽

Prime Concern ◽

Continuous Random Walk ◽

Transient Markov Chain ◽

Quasi Stationary Distribution

Let {Xn, n = 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩 + = {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix 𝐏 = (pij), and let pij(n) = P ∈ Xn = j, Xk ∈ 𝒩+ for k = 0, 1, ···, n | X0 = i], i, j 𝒩 +. The prime concern of this paper is conditions for the existence of the limits, qij say, of as n →∞. If the distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vector x = (xi) satisfying rxT = xT𝐏 and exists, where r is the convergence norm of 𝐏, i.e. r = R–1 and and T denotes transpose, then it is unique, positive elementwise, and qij(n) necessarily converge to xj as n →∞. Unlike existing results in the literature, our results can be applied even to the R-null and R-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix is R-transient is discussed to demonstrate the usefulness of our results.

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Some limit theorems for positive recurrent branching Markov chains: I

Advances in Applied Probability ◽

10.1017/s0001867800008557 ◽

1998 ◽

Vol 30 (03) ◽

pp. 693-710 ◽

Cited By ~ 3

Author(s):

Krishna B. Athreya ◽

Hye-Jeong Kang

Keyword(s):

Markov Chain ◽

Limit Theorems ◽

Law Of Large Numbers ◽

Large Deviation ◽

Position Distribution ◽

Discrete State ◽

Large Numbers ◽

Watson Process ◽

Positive Recurrent ◽

Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

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