Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T19:05:30.309Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get accessRights & Permissions [Opens in a new window]

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

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972, pp. 745 - 757
Copyright
Copyright © Applied Probability Trust 1972 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Benton, G. S. and Kahn, A. B. (1958) Spectra of large-scale atmospheric flow at 300 millibars. J. Met. 15, 404–410.2.0.CO;2>CrossRefGoogle Scholar
Brillinger, D. R. (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56, 375–390.Google Scholar
Brillinger, D. R. (1970) The frequency analysis of relations between stationary spatial series. Proceedings of the Twelfth Biennal Seminar on Time Series, Stochastic Processes, Convexity, Combinatorics. (Ed. Pike, R.) pp. 39–81. Canadian Mathematical Congress, Montréal.Google Scholar
Brillinger, D. R. (1972) The Frequency Analysis of Vector-Valued Time Series. Vol. 1. Holt, Rinehart and Winston, New York (to appear).Google Scholar
Hannan, E. J. (1963) The statistical analysis of hydrological time series. Proceedings of the National Symposium on Water Resources, Use and Management, pp. 233–243. Melbourne University Press.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
Loève, M. (1963). Probability Theory. 3rd Edition. Van Nostrand, Princeton.Google Scholar
Mann, H. B. and Wald, A. (1943) On stochastic limit and order relationships. Ann. Math. Statist. 14, 217–226.Google Scholar
Schoenberg, I. J. (1942) Positive definite functions on spheres. Duke Math. J. 9, 96–108.Google Scholar
Yaglom, A. M. (1961) Second order homogeneous random fields. Proc. Fourth Berkeley Symp. Vol. 2, 593–622.Google Scholar