AN L(2) CONVERGENCE THEOREM FOR RANDOM AFFINE MAPPINGS

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of the nth composition of these m aps in the Wasserstein metric via a contraction argument. The contract ion condition involves the operator norm of the expectation of a bilin ear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to c heck. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.