A Matrix Approach to Nonstationary Chains

A finite discrete nonstationary Markov chain is completely characterized (after the initial probability distribution has taken effect) by its time sequence of transition probability matrices P_i. The ith causative matrix C_i is defined as the product P_i⁻¹ (if it exists) times P_i+1. Thus, the causative matrices are analogous to derivatives in calculus as an indication of rate of change. The eigenvalues and eigenvectors of a constant causative matrix C have been found useful in their connection with the tendency of the chain to be convergent or divergent. Results for two-state chains are presented in some detail. A comprehensive bibliography of papers on non-stationary chains is included.