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Spectral analysis for a random process on the circle

Published online by Cambridge University Press:  14 July 2016

Roch Roy
Roch Roy*
Affiliation:
Université de Montréal
*
*Now at Université du Québec à Montréal.
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Abstract

A random process on the circle is a family of random variables X(P,t) indexed by the position P on the unit circle and by the time t (t = 0, + 1, ···). For a homogeneous and stationary process, using its Fourier series expansion, we deduce a spectral representation of the covariance function. The purpose of the paper is to develop a spectral analysis when X(P,t) is observed at all the points on the circle at t = 0, 1, ···, T – 1. The asymptotic distribution of the family of finite Fourier transforms, of the family of periodograms and of the family of spectral densities estimates are obtained from results available for vector-valued time series. Also, a sample covariance function for X (P,t) is defined and its asymptotic distribution derived.

Keywords

HOMOGENEOUS AND STATIONARY PROCESS ON THE CIRCLESPECTRAL REPRESENTATIONASYMPTOTIC DISTRIBUTIONPERIODOGRAMSSPECTRAL DENSITIES ESTIMATESSAMPLE COVARIANCE FUNCTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 745 - 757
Copyright
Copyright © Applied Probability Trust 1972 

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