Markov chain, ergodic

A homogeneous Markov chain with the following property: There are quantities (independent of )

(1)

where

are the transition probabilities. The distribution on the state space of the chain is called a stationary distribution: If for all , then for all and . A fundamental property of Markov chains,

enables one to find the without calculating the limits in (1).

Let

be the moment of first return to the state (for a discrete-time Markov chain), then

A similar (more complicated) relation holds for a continuous-time Markov chain.

The trajectories of an ergodic Markov chain satisfy the ergodic theorem: If is a function on the state space of the chain , then, in the discrete-time case,

while in the continuous-time case the sum on the left is replaced by an integral. A Markov chain for which there are and such that for all ,

(2)

is called geometrically ergodic. A sufficient condition for geometric ergodicity of an ergodic Markov chain is the Doeblin condition (see, for example, [1]), which for a discrete (finite or countable) Markov chain may be stated as follows: There are an and a state such that . If the Doeblin condition is satisfied, then for the constants in (2) the relation holds.

A necessary and sufficient condition for geometric ergodicity of a countable discrete-time Markov chain is the following (see [3]): There are numbers , and a finite set of states such that:

References

[1]	J.L. Doob, "Stochastic processes" , Wiley (1953)
[2]	K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967)
[3]	N.N. Popov, "Conditions for geometric ergodicity of countable Markov chains" Soviet Math. Dokl. , 18 : 3 (1977) pp. 676–679 Dokl. Akad. Nauk SSSR , 234 : 2 (1977) pp. 316–319

References

[a1]	D. Freedman, "Markov chains" , Holden-Day (1975)
[a2]	M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980)
[a3]	J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960)
[a4]	J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976)
[a5]	D. Revuz, "Markov chains" , North-Holland (1975)
[a6]	V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian)
[a7]	E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981)