On the ergodicity of slow-varying nonstationary Markov chains

We consider a nonstationary Markov chain: X(k + 1) = P(k)X(k), where X(k) = {x is an element of R(n) : x(i) >= 0 for all i = 1, 2, ..., n and Sigma(n)(i=1) x(i) = 1} and P(k) is a stochastic matrix for all k = 0, 1, 2,. The main purpose of this article is to present some new conditions to guarantee the ergodicity for slow-varying nonstationary Markov chains and the bound for variation of parallel to P(k + 1) - P(k)parallel to(1) to ensure the strongly ergodicity is constructed as well.