Numerical Bounds on Fluctuating Linear Processes

Abstract

Let A be an n×n primitive nonnegative matrix. The long-run behavior xk/‖xk‖1 of the linear process xk+1=xkA is determined by the stochastic eigenvector π of A. In this paper we consider the linear process xk+1=xkAk, where each Ak fluctuates about A, and provide numerical bounds on the difference between xk/‖xk‖ and π, thus showing how well π describes the long-run behavior of this fluctuating behavior.