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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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