The perturbation bound for the Perron vector of a transition probability tensor. Issue 6 (28th May 2013) - Li, Wen
SUMMARY 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. Copyright © 2013 John Wiley & Sons, Ltd.