The Fundamental Matrix, and Allied Topics

Abstract

In systems with a reasonably small number of components, where machine computation is feasible, techniques are needed to calculate explicitly the mean and variance of the passage time for the Markov chain N(t) from a state to a set B. When the entry set B^* is much smaller than B, the rank of the problem reduces to that of B^* provided that the transition probabilities p_mn(t) are known. In such calculations there is a fundamental matrix Z, the anologue of that in discrete time, whose properties are of interest. The fundamental matrix is related to the covariance function for stationary processes defined on the chain N(t). It also plays a key role in the central limit theorem for additive processes defined on the chain, in ergodic potential theory,^† and in perturbation theory for Markov chains.