State Space Decomposition for Large Markov Chains

1995 ◽  
pp. 587-590 ◽  
Author(s):  
Maria Rieders
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Space Decomposition

On a paper by Doeblin on non-homogeneous Markov chains

1981 ◽  
Vol 13 (02) ◽  
pp. 388-401
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Field Structure ◽  
Space Decomposition

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

On a paper by Doeblin on non-homogeneous Markov chains

1981 ◽  
Vol 13 (2) ◽  
pp. 388-401 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Field Structure ◽  
Space Decomposition

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

State Space Decomposition into Cyclic Subspaces and Transformation to the Jordan Form

1984 ◽  
Author(s):  
R. P. Guidorzi ◽  
T. E. Bullock ◽  
G. Basile
Keyword(s):  
State Space ◽  
Jordan Form ◽  
Space Decomposition

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

2017 ◽  
Vol 32 (4) ◽  
pp. 626-639 ◽  
Author(s):  
Zhiyan Shi ◽  
Pingping Zhong ◽  
Yan Fan
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Random Environment ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Countable State Space ◽  
Large Numbers ◽  
Countable State ◽  
Definition Of

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

R-theory for Markov chains on a topological state Space. II

1976 ◽  
Vol 34 (4) ◽  
pp. 269-278 ◽  
Author(s):  
David B. Pollard ◽  
Richard L. Tweedie
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Topological State

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).

Some sufficient conditions for non-ergodicity of markov chains

1985 ◽  
Vol 22 (01) ◽  
pp. 138-147 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Lyapunov Functions ◽  
Sufficient Conditions

Some sufficient conditions for non-ergodicity are given for a Markov chain with denumerable state space. These conditions generalize Foster's results, in that unbounded Lyapunov functions are considered. Our criteria directly extend the conditions obtained in Kaplan (1979), in the sense that a class of Lyapunov functions is studied. Applications are presented through some examples; in particular, sufficient conditions for non-ergodicity of a multidimensional Markov chain are given.

Reversibility of Markov chains with applications to storage models

1985 ◽  
Vol 22 (01) ◽  
pp. 123-137 ◽  
Author(s):  
Hideo Ōsawa
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Single Server ◽  
General State ◽  
Insurance Risk ◽  
General State Space ◽  
Necessary And Sufficient

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.

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