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 = (pi1,i2,...,im) with pi1,i2,...,im ≥ 0 and ∑ n i1=1 pi1 ,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 nonnegative and ∑i=1 xi = 1. The main contribution of this paper is to show that when P is perturbed to an another transition probability tensor P̃ by ∆P , the 1-norm error between x and x̃ is bounded by m, ∆P , 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 c © 2010 John Wiley & Sons, Ltd.