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August, 1973 The Central Limit Theorem for Random Motions of $d$-Dimensional Euclidean Space
Luis G. Gorostiza
Ann. Probab. 1(4): 603-612 (August, 1973). DOI: 10.1214/aop/1176996889


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.


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Luis G. Gorostiza. "The Central Limit Theorem for Random Motions of $d$-Dimensional Euclidean Space." Ann. Probab. 1 (4) 603 - 612, August, 1973.


Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0263.60010
MathSciNet: MR353408
Digital Object Identifier: 10.1214/aop/1176996889

Primary: 60F05
Secondary: 60K99

Keywords: central limit theorem , Haar measure , normal distribution , Random motions of Euclidean space

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 4 • August, 1973
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