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

Part of: Graph theory Probability theory on algebraic and topological structures Limit 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
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.

References

Baldi, P. and Rinott, Y. (1989) On normal approximations of distributions in terms of dependency graphs. Ann. Prob. 17, 1646–1650.Google Scholar
Baldi, P., Rinott, Y. and Stein, C. (1989) A normal approximation for the number of local maxima of a random function on a graph. Probability, Statistics and Mathematics: Papers in Honor of Samuel Karlin. ed. Anderson, T. W., Athreya, K. B. and Iglehart, D. L. Academic Press, New York.Google Scholar
Barbour, A. D. (1990) Stein's method for diffusion approximations. Prob. Theory Rel. Fields 84, 297–322.Google Scholar
Barbour, A. D., Holst, L. and Janson, J. (1992) Poisson Approximation. Clarendon, Oxford.Google Scholar
Barbour, A. D., Karoński, M. and Ruciński, A. (1989) A central limit theorem for decomposable random variables with applications to random graphs. J. Comb. Theory B 47, 125–145.Google Scholar
Bollobás, B. (1985) Random Graphs. Academic Press, New York.Google Scholar
Brewer, K. and Hanif, M. (1983) Sampling with Unequal Probabilities (Lecture Notes in Statistics 15). Springer, New York.Google Scholar
Cochran, W. (1977) Sampling Techniques. Wiley, New York.Google Scholar
Dall'Aglio, S., Kotz, S. and Salinetti, G., (eds) (1991) Advances in Probability Distributions with Given Marginals. Kluwer, Dordrecht.CrossRefGoogle Scholar
Dembo, A. and Rinott, Y. (1994) Some examples of normal approximations by Stein's method. IMA Conf. Proc. ed. Aldous, D. and Pemantle, R. (to appear).Google Scholar
Goldstein, L. and Rinott, Y. (1994) On multivariate normal approximations by Stein's method and size bias couplings. Technical Report. Google Scholar
Götze, F. (1991) On the rate of convergence in the multivariate CLT. Ann. Prob. 19, 724–739.Google Scholar
Horn, R. A. and Johnson, C. A. (1985) Matrix Analysis. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Janson, S. and Nowicki, K. (1991) The asymptotic distributions of generalized U-statistics with applications to random graphs. Prob. Theory Rel. Fields 90, 341–375.CrossRefGoogle Scholar
Karoński, M. and Ruciński, A. (1987) Poisson convergence of semi-induced properties of random graphs. Math. Proc. Camb. Phil. Soc. 101, 291–300.CrossRefGoogle Scholar
Kordecki, W. (1990) Normal approximation and isolated vertices in random graphs. In Random Graphs 1987. ed. Karoński, M., Jaworski, J. and Ruciński, A. Wiley, New York.Google Scholar
Luk, H. M. (1994) Stein's method for the Gamma distribution and related statistical applications. PhD dissertation. University of Southern California.Google Scholar
Palka, Z. (1984) On the number of vertices of a given degree in a random graph. J. Graph Theory 8, 167–170.Google Scholar
Reinert, G. (1994) A weak law of large numbers for empirical measures via Stein's method, and applications. PhD dissertation.CrossRefGoogle Scholar
Rinott, Y. (1995) On normal approximation rates for certain sums of dependent random variables. J. Appl. Comp. Math. (to appear).Google Scholar
Rinott, Y. and Rotar, V. (1994) A multivariate CLT for local dependence with n-1/2log n rate. J. Multivar. Anal. (to appear).Google Scholar
Stein, C. (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 583–602. University of California Press, Berkeley.Google Scholar
Stein, C. (1986) Approximate Computation of Expectations. IMS, Hayward, CA.Google Scholar
Stein, C. (1992) A way of using auxiliary randomization. Probability Theory, pp. 159–180. Walter de Gruyter, Berlin.Google Scholar