On some conditions of stability of Markov chains. (Russian) Zbl 0749.60070

Kozarovitskij, E. L.

Teor. Veroyatn. Mat. Stat., Kiev 44, 67-72 (1991).

Conditions of stability, of recurrence, and positive recurrence of discrete Markov chains are considered. The author investigates the necessity of the following condition introduced by R. L. Tweedie [Ann. Inst. Stat. Math. 32, No. 2, 283-290 (1980; Zbl 0452.60075)]: for the initial and perturbated Markov chains with transition matrices \((p_{ij})_{i,j\geq 0}\) and \((\tilde p_{ij})_{i,j\geq 0}\) there exists such a sequence \((\delta_ i)_{i\geq 0}\), \(\delta_ i\geq 0\), that \[ (1+\delta_ i)^{-1} p_{ij}\leq\tilde p_{ij}\leq(1+\delta_ i)p_{ij},\;i\neq j,\quad\text{and}\quad\prod_{i=0}^ \infty (1+\delta_ i)<\infty. \]

Reviewer: E.L.Kozarovitskij (Kiev)

MSC:

60J10

Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

stability; recurrence; positive recurrence; perturbated Markov chains

Citations:

Zbl 0452.60075