The perturbation bound for the Perron vector of a transition probability tensor

Abstract

In this paper, we study the perturbation bound for the Perron vector of an mth-order n-dimensional transition probability tensor P=(pi1,i2,...,im) with pi1,i2,...,im≥0 and ∑i1=1npi1,i2,...,im=1. The Perron vector x associated to the largest Z-eigenvalue 1 of P, satisfies Pxm-1=x where the entries xi of x are non-negative and ∑i=1nxi=1. The main contribution of this paper is to show that when P is perturbed to an another transition probability tensor P̃ by ΔP, the 1-norm error between x and x̃ is bounded by m, ΔP, and the computable quantity related to the uniqueness condition for the Perron vector x̃ of P̃. Based on our analysis, we can derive a new perturbation bound for the Perron vector of a transition probability matrix which refers to the case of m=2. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis. © 2013 John Wiley & Sons, Ltd.