ON THE GROWTH OF INFINITE PRODUCTS OF SLOWLY VARYING PRIMITIVE MATRICES

Let {A(t)} be a sequence of bounded nonnegative matrices [each with Perron root λ(t)] satisfying a condition of uniform primitivity and having nonzero entries bounded away from 0. In a generalization of a result on powers of primitive matrices it is shown that if the matrices A(t) vary slowly enough (i.e., {norm of matrix}A(t+1)-A(t){norm of matrix}≤ε for all t and some small ε), then, as soon as the entries pi,j(t) of the backward product A(t) A(t-1), ⋯, A(1) are positive, the ratios pi,j(t)/pi,j(t-1) are equal to λ(t)+hi,j(t), where for all t0 each hi,j(t0) is small in a precisely quantified manner: the larger t0 is and the slower the probability-normed Perron vectors V(t) of A(t) have changed in the recent past preceding t0, the smaller hi,j(t0) will be. (Other similar results are derived.) The resulting approximation pi,j(t)/pi,j(t-1)≈λ(t) is illustrated with two numerical examples. © 1991.