Limit theorems for functionals of ergodic Markov chains with general state space

1999 ◽  
Vol 139 (664) ◽  
pp. 0-0 ◽  
Author(s):  
Xia Chen
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Limit Theorems ◽  
General State ◽  
General State Space

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).

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.

Reversibility of Markov chains with applications to storage models

1985 ◽  
Vol 22 (1) ◽  
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.

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.

Numerical integration using V-uniformly ergodic Markov chains

2004 ◽  
Vol 41 (4) ◽  
pp. 1104-1112 ◽  
Author(s):  
Peter Mathé
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Numerical Integration ◽  
Error Bounds ◽  
General State ◽  
Transition Kernel ◽  
Interpolation Theory ◽  
Uniform Ergodicity ◽  
General State Space ◽  
Initial Distributions

We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.

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