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

1992 ◽  
Vol 29 (01) ◽  
pp. 21-36 ◽  
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.

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

1992 ◽  
Vol 29 (1) ◽  
pp. 21-36 ◽  
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.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (01) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

1965 ◽  
Vol 2 (1) ◽  
pp. 88-100 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Stationary Distributions ◽  
Transient States ◽  
Reverse Process ◽  
Finite Markov Chains ◽  
Quasi Stationary Distribution

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

1965 ◽  
Vol 2 (01) ◽  
pp. 88-100 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Stationary Distributions ◽  
Transient States ◽  
Reverse Process ◽  
Finite Markov Chains ◽  
Quasi Stationary Distribution

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (1) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distributionπof a Markov chain admits moments of the general form ∫f(x)π(dx), wherefis a general function; specific examples includef(x) =xrandf(x) =esx. In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

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.

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

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 timegivenXn=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 forA(J) (J →∞) are given for these examples.

Criteria for classifying general Markov chains

1976 ◽  
Vol 8 (04) ◽  
pp. 737-771 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
The State ◽  
General State ◽  
Classical Work ◽  
The Status ◽  
General State Space

The aim of this paper is to present a comprehensive set of criteria for classifying as recurrent, transient, null or positive the sets visited by a general state space Markov chain. When the chain is irreducible in some sense, these then provide criteria for classifying the chain itself, provided the sets considered actually reflect the status of the chain as a whole. The first part of the paper is concerned with the connections between various definitions of recurrence, transience, nullity and positivity for sets and for irreducible chains; here we also elaborate the idea of status sets for irreducible chains. In the second part we give our criteria for classifying sets. When the state space is countable, our results for recurrence, transience and positivity reduce to the classical work of Foster (1953); for continuous-valued chains they extend results of Lamperti (1960), (1963); for general spaces the positivity and recurrence criteria strengthen those of Tweedie (1975b).

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