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, ∞).

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 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.

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}} .

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.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

Semi-Markov Replacement Chains

1994 ◽  
Vol 26 (3) ◽  
pp. 728-755 ◽  
Author(s):  
Ioannis I. Gerontidis
Keyword(s):  
Markov Chain ◽  
Control Theory ◽  
Markov Chains ◽  
Time Dependent ◽  
Applied Probability ◽  
Probability Models ◽  
Natural Manner ◽  
Environmental Behaviour ◽  
New Process

We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.

Markov Chains in Many Dimensions

1994 ◽  
Vol 26 (3) ◽  
pp. 756-774 ◽  
Author(s):  
Dimitris N. Politis
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Maximum Entropy ◽  
Stationary Distribution ◽  
Random Fields ◽  
Special Class ◽  
Markov Random Fields ◽  
One Dimension ◽  
Stationary Markov Chain ◽  
Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

scholarly journals Assessing significance in a Markov chain without mixing

2017 ◽  
Vol 114 (11) ◽  
pp. 2860-2864 ◽  
Author(s):  
Maria Chikina ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Stationary Distribution ◽  
Potential Application ◽  
Null Hypothesis ◽  
Value Function ◽  
Mixing Time ◽  
Statistical Test ◽  
Reversible Markov Chain ◽  
A Value

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p=2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p≈ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.

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