On the Convergence of the Discrete-Time Homogeneous Markov Chain

The evolution of a discrete-time Markov Chain (MC) is determined by the evolution equation p(T)(t) = p(T)(t - 1).P, where p(t) stands for the stochastic state vector at time t, t is an element of N, P interprets the stochastic transition matrix of the MC, and the superscript (T) denotes transposition of the respective column vector (or matrix). The present chapter examines under which conditions concerning the stochastic matrix P, a set of stochastic vectors, {p(t - 1)}, representing a hypersphere on the set of the attainable structures of the MC, is transformed into a stochastic set {p(t)} also representing a hypersphere of the MC. The results concerning the form of the transition matrix P are derived by means of the product PPT. The set of the matrices P turns out to be a subset of the set of the doubly stochastic matrices.