Markov chain properties in terms of column sums of the transition matrix

TL;DR

This paper investigates how the column sums of a stochastic matrix influence key properties of the associated Markov chain, introducing new relationships and inequalities using a novel generalized matrix inverse.

Contribution

It presents new relationships and inequalities linking column sums of transition matrices to Markov chain properties, using a previously unconsidered generalized inverse.

Findings

Derived inequalities relating column sums to stationary distribution

Established connections between column sums and mean first passage times

Provided partial answers to how column sums affect the Kemeny constant

Abstract

Questions are posed regarding the influence that the column sums of the transition probabilities of a stochastic matrix (with row sums all one) have on the stationary distribution, the mean first passage times and the Kemeny constant of the associated irreducible discrete time Markov chain. Some new relationships, including some inequalities, and partial answers to the questions, are given using a special generalized matrix inverse that has not previously been considered in the literature on Markov chains.