Markov Chains with Stationary Transition Probabilities

1961 ◽  
Vol 45 (354) ◽  
pp. 362
Author(s):  
D. G. Kendall ◽  
Kai Lai Chung
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Stationary Transition

Approximating kth-order two-state Markov chains

1992 ◽  
Vol 29 (4) ◽  
pp. 861-868 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Upper Bound ◽  
Transition Probabilities ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Stationary Transition

In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (01) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L 2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

Approximating kth-order two-state Markov chains

1992 ◽  
Vol 29 (04) ◽  
pp. 861-868 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Upper Bound ◽  
Transition Probabilities ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Stationary Transition

In this paper, we consider kth-order two-state Markov chains {Xi } with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σ x |𝐏(Sn = x) − 𝐏(Y = x)|, where S n = X 1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (1) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

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