Improved bounds for a condition number for Markov chains

Abstract

Let P be the transition matrix for an n-state, homogeneous, ergodic Markov chain. Set Q= I− P and let Q #=[ q # i, j ] be the group (generalized) inverse of Q. A well-known condition number, due to Funderlic and Meyer, which is used in the error analysis for the computation of the stationary distribution vector π=[ π 1,…, π n ] T of the chain, is κ 4:=max 1⩽ i, j⩽ n | q # i, j |. In this paper we refine two upper estimates on κ 4 due to Meyer. In the course of proving one of our results we show that | q # i, j |⩽ π j (1− π i )∥ Q −1 j ∥ ∞, where Q j is the ( n−1)×( n−1) principal submatrix of Q obtained from deleting its jth row and column, and we characterize the case of equality. The fact that we have a tight upper bound on the individual entries of the group inverse allows us to apply it in other contexts in which the group inverse arises. For an irreducible nonnegative matrix, such applications include, for instance, bounds on the second order partial derivative of the Perron root with respect to any entry of the matrix and on the elasticity of the Perron root with respect to any entry of the matrix.