Numerical Methods for Computing Stationary Distributions of Finite Irreducible Markov Chains

2000 ◽  
pp. 81-111 ◽  
Author(s):  
William J. Stewart
Keyword(s):  
Numerical Methods ◽  
Markov Chains ◽  
Stationary Distributions

Numerical Methods in Markov Chains and Bulk Queues

1972 ◽  
Author(s):  
Tapan P. Bagchi ◽  
James G. C. Templeton
Keyword(s):  
Numerical Methods ◽  
Markov Chains

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

1966 ◽  
Vol 3 (02) ◽  
pp. 403-434 ◽  
Author(s):  
E. Seneta ◽  
D. Vere-Jones
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Recent Work ◽  
Discrete Time ◽  
Branching Process ◽  
Stationary Distributions

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

A Remark on the Stability of the Stationary Distributions of MARKOV Chains

1977 ◽  
Vol 78 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Matthias Kotzurek ◽  
Dietrich Stoyan
Keyword(s):  
Markov Chains ◽  
Stationary Distributions ◽  
The Stability

Moments for stationary and quasi-stationary distributions of markov chains

1985 ◽  
Vol 22 (01) ◽  
pp. 148-155 ◽  
Author(s):  
E. Seneta ◽  
R. L. Tweedie
Keyword(s):  
Invariant Measure ◽  
Markov Chains ◽  
Stationary Distributions ◽  
Limit Distributions ◽  
Sufficient Set ◽  
Watson Process ◽  
Necessary And Sufficient ◽  
Conditional Limit ◽  
Finite Moments

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

Quasi-stationary distributions for Markov chains on a general state space

1974 ◽  
Vol 11 (4) ◽  
pp. 726-741 ◽  
Author(s):  
Richard. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
State Space ◽  
Finiteness Condition ◽  
General State ◽  
Stationary Distributions ◽  
General State Space ◽  
Countably Infinite ◽  
Stationary Behaviour

The quasi-stationary behaviour of a Markov chain which is φ-irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspace is finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.

scholarly journals A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zhao-Li Shen ◽  
Ting-Zhu Huang ◽  
Bruno Carpentieri ◽  
Chun Wen
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Numerical Experiments ◽  
Multigrid Method ◽  
Cost Effective ◽  
Multigrid Methods ◽  
Stationary Distributions ◽  
Convergence Behavior ◽  
Aggregation Procedure ◽  
Smoothed Aggregation

Smoothed aggregation multigrid method is considered for computing stationary distributions of Markov chains. A judgement which determines whether to implement the whole aggregation procedure is proposed. Through this strategy, a large amount of time in the aggregation procedure is saved without affecting the convergence behavior. Besides this, we explain the shortage and irrationality of the Neighborhood-Based aggregation which is commonly used in multigrid methods. Then a modified version is presented to remedy and improve it. Numerical experiments on some typical Markov chain problems are reported to illustrate the performance of these methods.

Computing limiting stationary distributions of small noisy networks

2002 ◽  
Vol 39 (1) ◽  
pp. 161-178
Author(s):  
Fred Richman ◽  
Katarzyna Winkowska-Nowak
Keyword(s):  
Neural Network ◽  
Markov Chains ◽  
Time Scales ◽  
Nonnegative Matrix ◽  
Independent Interest ◽  
Stationary Distributions ◽  
Mathematical Ideas

The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest.

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