scholarly journals The Condition of Regularity in Simple Markov Chains

1956 ◽  
Vol 9 (3) ◽  
pp. 387
Author(s):  
J Gani
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Latent Root ◽  
Necessary And Sufficient Conditions ◽  
A Chain ◽  
Necessary And Sufficient

The paper generalizes a proof, and outlines an alternative to it, for the well-known theorem on the conditions of regularity in a simple Markov chain; this is that the necessary and sufficient conditions for a chain to be regular are that the latent root 1 of the stochastic matrix for the chain must be simple, and the remaining roots have moduli less than 1.

scholarly journals Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities

1978 ◽  
Vol 15 (4) ◽  
pp. 848-851 ◽  
Author(s):  
Jean-François Mertens ◽  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Of A Solution ◽  
Necessary And Sufficient ◽  
Irreducible Markov Chain

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk < yo, · · ·, yN–1 for some k ≧ N, is necessary and sufficient for transience.

Limiting second moments for transient states of Markov chains

1973 ◽  
Vol 10 (04) ◽  
pp. 891-894
Author(s):  
H. P. Wynn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Time Points ◽  
Transient States ◽  
Second Moments ◽  
Necessary And Sufficient

The set of transient states of a Markov chain is considered as a system. If numbers of arrivals to the system at discrete time points have constant mean and covariance matrix then there is a limiting distribution of numbers in the states. Necessary and sufficient conditions are given for this distribution to yield zero correlations between states.

Transience and recurrence of an interesting Markov chain

1974 ◽  
Vol 11 (04) ◽  
pp. 818-824 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chain ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Recurrence ◽  
Null Recurrence ◽  
A Chain ◽  
Necessary And Sufficient ◽  
Countable Case

This note studies the natural extension to the countable case of a chain considered by Hendricks (1972) and gives necessary and sufficient conditions for transience, null recurrence and positive recurrence.

The embedding problem for Markov chains with three states

1980 ◽  
Vol 87 (2) ◽  
pp. 285-294 ◽  
Author(s):  
Halina Frydman
Keyword(s):  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Parameter Family ◽  
Necessary And Sufficient Conditions ◽  
Embedding Problem ◽  
Stochastic Matrices ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Two Parameter

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matricessatisfyingand such thatThough extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

Transience and recurrence of an interesting Markov chain

1974 ◽  
Vol 11 (4) ◽  
pp. 818-824 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chain ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Recurrence ◽  
Null Recurrence ◽  
A Chain ◽  
Necessary And Sufficient ◽  
Countable Case

This note studies the natural extension to the countable case of a chain considered by Hendricks (1972) and gives necessary and sufficient conditions for transience, null recurrence and positive recurrence.

scholarly journals Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities

1978 ◽  
Vol 15 (04) ◽  
pp. 848-851 ◽  
Author(s):  
Jean-François Mertens ◽  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Of A Solution ◽  
Necessary And Sufficient ◽  
Irreducible Markov Chain

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk &lt; y o, · · ·, yN –1 for some k ≧ N, is necessary and sufficient for transience.

Limiting second moments for transient states of Markov chains

1973 ◽  
Vol 10 (4) ◽  
pp. 891-894 ◽  
Author(s):  
H. P. Wynn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Time Points ◽  
Transient States ◽  
Second Moments ◽  
Necessary And Sufficient

The set of transient states of a Markov chain is considered as a system. If numbers of arrivals to the system at discrete time points have constant mean and covariance matrix then there is a limiting distribution of numbers in the states. Necessary and sufficient conditions are given for this distribution to yield zero correlations between states.

Lumpability and marginalisability for continuous-time Markov chains

1993 ◽  
Vol 30 (3) ◽  
pp. 518-528 ◽  
Author(s):  
Frank Ball ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Ion Channel ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Continuous Time Markov Chains ◽  
Probabilistic Proof ◽  
Necessary And Sufficient ◽  
Mutually Independent

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

Poisson theorem for non-homogeneous Markov chains

1989 ◽  
Vol 26 (03) ◽  
pp. 637-642 ◽  
Author(s):  
Janusz Pawłowski
Keyword(s):  
Markov Chains ◽  
Poisson Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compound Poisson Distribution ◽  
Convergence In Distribution ◽  
Compound Poisson ◽  
Necessary And Sufficient

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

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