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The convolution metric dg

Published online by Cambridge University Press:  24 October 2008

J. E. Yukich
J. E. Yukich
Affiliation:
Université Louis Pasteur, Strasbourg, France
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Abstract

Summary

We introduce and study a new metric on denned by

where is the space of probability measures on ℝk and where g: ℝk→ is a probability density satisfying certain mild conditions. The metric dg, relatively easy to compute, is shown to have useful and interesting properties not enjoyed by some other metrics on . In particular, letting pn denote the nth empirical measure for P, it is shown that under appropriate conditions satisfies a compact law of the iterated logarithm, converges in probability to the supremum of a Gaussian process, and has a useful stochastic integral representation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 533 - 540
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Billingsley, P.. Convergence of Probability Measures. (Wiley, 1968).Google Scholar
[2]Dudley, R. M.. Convergence of Baire measures. Studio Math. 27 (1966), 251–268.CrossRefGoogle Scholar
[3]Dudley, R. M.. The speed of mean Glivenko-Cantelli convergence. Annals of Math. Stat. (1) 40 (1969), 40–50.CrossRefGoogle Scholar
[4]Dudley, R. M.. Central limit theorems for empirical measures. Ann. Probab. 6 (1978), 899–929, correction, 7 (1979), 909–911.CrossRefGoogle Scholar
[5]Dudley, R. M.. Donsker classes of functions. Statistics and Related Topics (Proc. Symp. Ottawa, 1980) (North Holland, 1981), 341–352.Google Scholar
[6]Dudley, R. M.. A Course on Empirical Processes. Lecture Notes in Mathematics, no. 1097 (Springer-Verlag, 1984).Google Scholar
[7]Dudley, R. M. and Philipp, W.. Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrschein. 62 (1983), 509–552.CrossRefGoogle Scholar
[8]Feller, W.. An Introduction to Probability Theory and Its Applications (Wiley, 1968).Google Scholar
[9]Fortet, R. and Mourier, E.. Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Norm. Sup. 70 (1953), 266–285.Google Scholar
[10]Jain, N. C. and Marcus, M. B.. Central limit theorems for C(S) valued random variables. J. Funct. Anal. 19 (1975), 216–231.CrossRefGoogle Scholar
[11]Kuelbs, J. and Dudley, R. M.. Log log laws for empirical measures. Ann. Probab. 8 (1980), 405–418.CrossRefGoogle Scholar
[12]Rudin, W.. Fourier Analysis on Groups (Interscience Publishers, 1962).Google Scholar
[13]Zolotarev, V. M.. Probability metrics. Theory Probab. Appl. 28 (2) (1984), 278–302.CrossRefGoogle Scholar
[14]Zuker, M.. Speeds of convergence of random probability measures. Ph.D. Dissertation, M.I.T. (1974).Google Scholar