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On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

Part of: Parametric inference Nontrigonometric harmonic analysis Stochastic processes Inference from stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Sergio Albeverio  and
Shuji Kawasaki
Sergio Albeverio*
Affiliation:
Universität Bonn
Shuji Kawasaki*
Affiliation:
Hitotsubashi University
*
∗ Postal address: Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany. Email address: albeverio@uni-bonn.de
∗∗ Current address: Center for Tsukuba Advanced Research Alliance, The University of Tsukuba, Tennoudai 1-1-1, Tsukuba-shi, Ibaraki-ken, 305-8577, Japan. Email address: kawasaki@wslab.risk.tsukuba.ac.jp
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Abstract

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We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

Keywords

Wavelet coefficientlocalizationparameter estimationprocess with stationary incrementspositive-definite functionlikelihoodnoncentral limit theoremcentral limit theoremlong-range dependence

MSC classification

Primary: 42C40: Wavelets and other special systems 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G18: Self-similar processes
Secondary: 60G15: Gaussian processes 62F10: Point estimation 62M10: Time series, auto-correlation, regression, etc.
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 938 - 962
Copyright
Copyright © Applied Probability Trust 2005 

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