Asymptotic behaviour under iterated random linear transformations

Abstract

Let V be a finite-dimensional real vector space. We describe conditions for a sequence of the form {μⁱ∗ν}, where μ is a probability measure on GL(V ) (μⁱ denotes the i-th convolution power of μ) and ν is a finite positive measure on V, to converge in distribution (in the vague topology) to the zero measure on V. The conditions depend on μ only via the closed subgroup of GL( V ) generated by the support of μ.