Asymptotic behaviour under iterated random linear transformations

Let $V$ be a finite-dimensional real vector space. We describe conditions for a sequence of the form $\{\mu^i*\nu\}$, where $\mu$ is a probability measure on ${\mathop{\rm GL}\nolimits}(V)$ ($\mu^i$ denotes the $i$-th convolution power of $\mu$) and $\nu$ 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 $\mu$ only via the closed subgroup of ${\mathop{\rm GL}\nolimits}(V)$ generated by the support of $\mu$.

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Published 1 January 2004