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Multivariate normal approximations by Stein's method and size bias couplings

Part of:Probability theory on algebraic and topological structuresGraph theoryLimit theorems

Published online by Cambridge University Press:  14 July 2016

Larry Goldstein and
Yosef Rinott
Larry Goldstein*
Affiliation:
University of Southern California
Yosef Rinott*
Affiliation:
University of California, San Diego
*
∗ Postal address: Department of Mathematics DRB-155, University of Southern California, Los Angeles, CA 90089–1113, USA.
∗∗ Postal address: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA.
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Abstract

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

Keywords

STEIN'S METHODCOUPLINGSIZE BIASRANDOM GRAPHSMULTIVARIATE CENTRAL LIMIT THEOREMS

MSC classification

Primary:60F05: Central limit and other weak theorems
Secondary:60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)05C80: Random graphs

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996, pp. 1 - 17
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This work was supported in part by NSF grants DMS 90–05833 and DMS 95–05075.

This work was supported in part by NSF grant DMS 92–05759.

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