On the sequence of powers of a stochastic matrix with large exponent Original Research Article

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least left floor(n2−2n+2)/2right floor+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λn−αλn−j−(1−α)λn−k. In this paper, we prove that for any mgreater-or-equal, slantedn, if αless-than-or-equals, slant1/2, then short parallelAm+k−Amshort parallel∞less-than-or-equals, slantshort parallelAm−1wTshort parallel∞, where 1 is the all-ones vector and wT is the left-Perron vector for A, normalized so that wT1=1. We also prove that if jgreater-or-equal, slantedn/2, ngreater-or-equal, slanted31 and image, then short parallelAm+j−Amshort parallel∞less-than-or-equals, slantshort parallelAm−1wTshort parallel∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence Am.