Products of random mappings

Suppose that X-1, X-2,... is a stationary stochastic process of positive k x k matrices, and let Y-n(1) = (XXn-1)-X-n ...X-1 be the corresponding prod uct matrices. For a special case, Bellman showed that the elements [Y-n(1)] (ij) converge in the sense that n(-1)E{log[Y-n(1)](ij)} --> a as n --> infi nity. The constant a is independent of i and j. Bellman also conjectured th at, asymptotically, the n(-1/2){log[Y-n(1)](ij) - na} terms are distributed according to a normal distribution with a common variance, independent of ij. Later Furstenberg and Kesten generalized and strengthened Bellman's res ult and established the validity of his conjecture. This paper extends these results to the case of nonlinear mappings that are monotonic and homogeneous of degree one on R-+(k). Specifically, given a s tationary process H-1, H-2,... of such mappings, we define the composite ma ppings F-n(1)(.) = H-n (Hn-1(...(H-1(.)...). Under appropriate conditions, the components [F-n(1)(x(0))](i) have the property that, almost surely, n(- 1) log[F-n(1)(x(0))](i) --> a independent of x(0) and i. Furthermore the co mponents n(-1/2){log[F-n(1)(x(0))](i) - na} are asymptotically distributed according to a normal distribution with a common variance.