An L 2 convergence theorem for random affine mappings

Abstract

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