On Random Maps Correlated With Random Densities

Abstract

Let q = (q ij ) : 1 ≤ i ≤ I, 1 ≤ j ≤ J be a bivariate probability vector, let T = (T 1 , • • • , T I ) be a sequence of ρ-nonsingular transformations defined on a probability space (E, B, ρ) and let f" = (f 1 , • • • , f J ) be a sequence of densities in L 1 (ρ). In this paper, we construct in a natural way, a discrete random dynamical system (with skew product Φ) generated by T and the first marginal of q and a random density ξ generated by f" and the second marginal of q. Moreover, we characterize the Φ-invariance of ξ.