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The order of the approximation to a Wiener process by its Fourier series

Published online by Cambridge University Press:  24 October 2008

Peter Hall
Peter Hall
Affiliation:
Australian National University
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Abstract

Let F be a distribution function with support confined to ( − π, π), W a Wiener process on (0, 1), and W0 a Brownian bridge. We examine the order of the approximation to W(F) and W0(F) by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 547 - 562
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

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