Abstract
Fix a probability measure on the space of isometries of Euclidean space $\mathbf{R}^d$. Let $Y_0=0,Y_1,Y_2,\ldots\in\mathbf{R}^d$ be a sequence of random points such that $Y_{l+1}$ is the image of $Y_l$ under a random isometry of the previously fixed probability law, which is independent of $Y_l$. We prove a Local Limit Theorem for $Y_l$ under necessary nondegeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of $Y_l$ on scales $e^{-cl^{1/4}}
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