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

In this paper, we study the perturbation bound for the Perron vector of an mth-order n-dimensional transition probability tensor P = (pi(1), i(2), ... , i(m)) with pi(1), i(2), ... , i(m) >= 0 and Sigma(n)(i1=1) pi(1), i(2), ... , i(m) = 1. The Perron vector x associated to the largest Z-eigenvalue 1 of P, satisfies Px(m-1) = x where the entries x(i) of x are non-negative and Sigma(n)(i=1) x(i) = 1. The main contribution of this paper is to show that when P is perturbed to an another transition probability tensor (P) over tilde by Delta P, the 1-norm error between x and (x) over tilde is bounded by m, Delta P, and the computable quantity related to the uniqueness condition for the Perron vector (x) over tilde of (P) over tilde. 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. Copyright (C) 2013 John Wiley & Sons, Ltd.