The Central Limit Theorem for Random Motions of $d$-Dimensional Euclidean Space

Abstract

Let $g_1, g_2,\cdots$ be random elements of the Euclidean group of motions of $d$-dimensional Euclidean space $R^d (d \geqq 1)$, that are independent and identically distributed. The product $g_1\cdots g_n$ is represented in the form $t(n)r(n)$, where $t(n)$ is a translation and $r(n)$ is a rotation. In this paper it is shown that under natural conditions $r(n)$ and $n^{-\frac{1}{2}}t(n)$ jointly converge weakly as $n \rightarrow \infty$ to the product distribution of the Haar measure on a certain closed subgroup of the rotations group, and a normal distribution on $R^d$, with mean zero and covariance matrix $\sigma^2\mathbf{I}$ ($\mathbf{I}$ is the identity matrix), and the value of $\sigma^2$ is identified.