On a Markov Chain with Denumerable Number of States and Transition Probabilities Dependent on Probability States

Abstract

The authors consider homogeneous Markov chain ξt, t ≥ 0 with a denumerable number of states and transition probabilities dependent on the states of that chain. If the chain ξt, t ≥ 0 is assumed to be ergodic for stationary distribution {p ± k } , k ≥ 0 , it is established that a unique solution to the differential equations system relative to the generating functions P ± (θ) , |θ| ≤ 1 of that distribution { p ± k } , k ≥ 0 exists. This condition is found in the form of the inequality ∥G∥ ≤ e 2 . It is based on Fubini’s theorem from the theory of functions and on the existence of the bound G ≡ G∞ = Gn = limn→∞ Eeθ−η , Eis the identity matrix. Using the principle of the matrix theory by induction, we get that