Generalized inverses of Markovian kernels in terms of properties of the Markov chain

TL;DR

This paper characterizes all one-condition generalized inverses of the Markovian kernel I - P for finite irreducible Markov chains, linking them to stationary probabilities, mean first passage times, and second moments, with applications to Kemeny's constant.

Contribution

It provides a unified characterization of generalized inverses of I - P in terms of Markov chain properties, including special sub-families and their applications.

Findings

All such inverses are uniquely specified by stationary probabilities and mean first passage times.

Sub-families include the group inverse, fundamental matrix, and Moore-Penrose g-inverse.

Applications to Kemeny's constant and chain perturbations are demonstrated.

Abstract

All one-condition generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include the group inverse of I - P, Kemeny and Snell's fundamental matrix of the Markov chain and the Moore-Penrose g-inverse. The elements of some sub-families of the generalized inverses can also be re-expressed involving the second moments of the recurrence time variables. Some applications to Kemeny's constant and perturbations of Markov chains are also considered.