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June, 1975 The Multivariate Central Limit Theorem for Regular Convex Sets
T. K. Matthes
Ann. Probab. 3(3): 503-515 (June, 1975). DOI: 10.1214/aop/1176996356

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random vectors in $R_k$. Let $P_n$ denote the probability measure induced by the normalized sum and let $Q_n$ denote the multivariate Edgeworth signed measure with terms through $n^{-\frac{1}{2}}$. If $C$ is a member of a class of convex bodies whose boundaries are sufficiently smooth and possess positive Gaussian curvatures, and $X_1$ has fourth moments, it is shown that $P_n(C) - Q_n(C) = 0(n^{-k/(k + 1)})$ where the bound is uniform. If, moreover, $X_1$ has a nonlattice distribution, the difference is $o(n^{-k/(k + 1)})$.

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T. K. Matthes. "The Multivariate Central Limit Theorem for Regular Convex Sets." Ann. Probab. 3 (3) 503 - 515, June, 1975. https://doi.org/10.1214/aop/1176996356

Information

Published: June, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0347.60023
MathSciNet: MR375434
Digital Object Identifier: 10.1214/aop/1176996356

Subjects:
Primary: 62E20
Secondary: 60B15 , 60F05

Keywords: Edgeworth expansion , multivariate central limit theorem , nonlattice distributions , probability of convex sets , rates of convergence

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 3 • June, 1975
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