In this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.