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 Pi. The ith causative matrix Ci is defined as the product Pi−1 (if it exists) times Pi+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.