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

SUMMARY In this paper, we study the perturbation bound for the Perron vector of an m th‐order n ‐dimensional transition probability tensor P = (pi 1 , i 2 , … , i m) withpi 1 , i 2 , … , i m⩾ 0 and∑i 1 = 1 npi 1 , i 2 , … , i m= 1 . The Perron vector x associated to the largest Z ‐eigenvalue 1 of P , satisfies P x m − 1 = x where the entries x i of x are non‐negative and∑ i = 1 nx i = 1 . The main contribution of this paper is to show that when P is perturbed to an another transition probability tensorP ̃ by Δ P , the 1‐norm error between x andx ̃ is bounded by m , Δ P , and the computable quantity related to the uniqueness condition for the Perron vectorx ̃ ofP ̃ . 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 © 2013 John Wiley & Sons, Ltd.