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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

REFERENCES

(1)Cramér, H. and Leadbetter, M. R.Stationary and related stochastic processes (Wiley, New York, 1967).Google Scholar
(2)Doob, J. L.Stochastic processes (Wiley, New York, 1953).Google Scholar
(3)Eagleson, G. K.Some simple conditions for limit theorems to be mixing. Teor. Verajnost. i Primen 21 (1977), 653–660.Google Scholar
(4)Garsia, A. M. Continuity properties of Gaussian processes with multi-dimensional time parameter. Proc. Sixth Berkeley Symposium Math. Statist. Probability, vol. II (University of California Press, 1972), pp. 369–374.Google Scholar
(5)Komlós, J., Major, P. and Tusnády, G.An approximation of partial sums of independent rv's, and the sample df, I. Z. Wahrscheinlichkeitstheorie verw. Geb. 32 (1975), 111–131.CrossRefGoogle Scholar
(6)Kronmal, R. and Tarter, M.The estimation of probability densities and cumulatives by Fourier series methods. J. Amer. Statist. Assoc. 63 (1968), 925–952.Google Scholar
(7)Loeve, M.Probability Theory, 3rd edn (Van Nostrand, Princeton, N.J., 1963).Google Scholar
(8)Silverman, B. W.On a Gaussian process related to multivariate probability density estimation. Math. Proc. Camb. Phil. Soc. 80 (1976), 135–144.CrossRefGoogle Scholar
(9)Silverman, B. W.Wreak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 (1978), 177–184.CrossRefGoogle Scholar
(10)Silverman, B. W.Choosing the window width when estimating a density. Biometrika 65 (1978), 1–11.CrossRefGoogle Scholar
(11)Tarter, M. E. and Kronmal, R. A.Estimation of the cumulative by Fourier series methods and application to the insertion problem. Proc. ACM 23 (1968), 491–497.Google Scholar
(12)Whittaker, E. T. and Watson, G. N.Modern analysis, 4th edn (Cambridge University Press, Cambridge, 1927).Google Scholar
(13)Zygmund, A.Trigonometric series, vol. I, 2nd edn (Cambridge University Press, Cambridge, 1959).Google Scholar