On sociologically derived mathematics

Abstract

Motivated by empirical studies of labor mobility, in 1970 Sy Spilerman approached me with a question about when a given stochastic matrix, $P$, can be reached from the identity matrix, $I$, in a prescribed period of time via the Kolmogorov forward and backward differential equations. A complete answer to this question for time-homogeneous Markov chains was provided in 1976—summarized herein—and a diversity of time-inhomogeneous cases have been studied since then and are reviewed in the present paper. Of particular interest for the future would be solutions for this reachability/embedding problem for mixtures of Markov chains. Also for the future would be studies of partial identifiability of mixtures of Markov chains when their observed transitions do not satisfy reachability/embedding conditions.