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Functional limit theory for the spectral covariance estimator

Part of:Nonparametric inferenceLimit theorems

Published online by Cambridge University Press:  14 July 2016

Dominique Dehay and
Jacek Leśkow
Dominique Dehay*
Affiliation:
Université de Rennes 1
Jacek Leśkow*
Affiliation:
University of California, Santa Barbara
*
∗ Postal address: IRMAR, Université de Rennes 1, Campus Beaulieu, 35042 Rennes, France.
∗∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106–3110, USA.
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Abstract

Processes that exhibit repeatability in their kth-order moments are frequently studied in signal analysis. Such repeatability can be conveniently expressed with the help of almost periodicity. In particular, almost periodically correlated (APC) processes play an important role in the analysis of repeatable signals. This paper presents a study of asymptotic distributions of the estimator of the spectral covariance function for APC processes. It is demonstrated that, for a large class of APC processes, the functional central limit theorem holds.

Keywords

ALMOST PERIODIC PROCESSFUNCTIONAL CENTRAL LIMIT THEOREMMIXING PROPERTY

MSC classification

Primary:60F17: Functional limit theorems; invariance principles
Secondary:62G20: Asymptotic properties

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996, pp. 1077 - 1092
Copyright
Copyright © Applied Probability Trust 1996 

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References

Amerio, L. and Prouse, G. (1971) Almost Periodic Functions and Functional Equations. Van Nostrand, Reinhold, New York.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Dehay, D. (1991) On the product of two harmonizable processes. Stoch. Proc. Appl. 38, 347–358.CrossRefGoogle Scholar
Dehay, D. (1995) Asymptotic behavior of estimators of cyclic functional parameter for some nonstationary processes. Statist. Decisions 13, 273–286.Google Scholar
Dehay, D. and Leskow, J. (1994) Frequency determination for almost periodically correlated stochastic processes. Preprint. Google Scholar
Feder, M. (1993) Parameter estimation and extraction of helicopter signals observed with a wide-band interference. IEEE Trans. Signal Processing SP-41, 232–244.CrossRefGoogle Scholar
Gardner, W. A. (1985) Introduction to Random Processes with Applications to Signals and Systems. Macmillan, New York.Google Scholar
Gardner, W. A. (1991) Exploitation of spectral redundancy in cyclostationary signals. IEEE Signal Processing Mag. 8, 14–36.CrossRefGoogle Scholar
Gerr, N. L. and Hurd, H. L. (1991) Graphical methods for determining the presence of periodic correlation. J. Time Series Anal. 12, 337–350.Google Scholar
Gladyshev, E. (1963) Periodically and almost periodically correlated random processes with continuous time parameter. Theory Prob. Appl. 8, 173–177.CrossRefGoogle Scholar
Goodman, N. R. (1965) Statistical tests for stationarity within the framework of harmonizable processes. Research Report AD619270. Rockeydyne, Canoga Park, CA.Google Scholar
Hannan, E. J. (1973) The estimation of frequency. J. Appl. Prob. 10, 515–519.CrossRefGoogle Scholar
Hurd, H. L. (1989) Nonparametric time series analysis for periodically correlated processes. IEEE Trans. Inform. Theory IT-35, 350–359.CrossRefGoogle Scholar
Hurd, H. L. (1991) Correlation theory for the almost periodically correlated processes with continuous time parameter. J. Multivar. Anal. 37, 24–45.CrossRefGoogle Scholar
Hurd, H. L. and Leskow, J. (1992a) Estimation of the Fourier coefficient functions and their spectral densities for f-mixing almost periodically correlated processes. Statist. Prob. Lett. 14, 299–306.CrossRefGoogle Scholar
Hurd, H. L. and Leskow, J. (1992b) Strongly consistent and asymptotically normal estimation of the covariance for almost periodically correlated processes. Statist. Decisions 10, 201–225.Google Scholar
Martin, D. E. (1990) Estimation of the minimal period of periodically correlated processes. PhD thesis. University of Maryland.Google Scholar
McLeish, D. L. (1975) Invariance principle for dependent variables. Z. Wahrscheinlichkeitsth. 32, 165–178.CrossRefGoogle Scholar
Rabiner, L. R. (1977) On the use of autocorrelation analysis for pitch detection. IEEE Trans. Acoust. Speech. Signal Processing ASSSP-25, 24–33.CrossRefGoogle Scholar
Tian, C. J. (1988) A limiting property of sample autocovariances of periodically correlated processes with application to period determination. J. Time Series Anal. 9, 411–417.CrossRefGoogle Scholar
Vecchia, A. V. (1985) Maximum likelihood estimation for periodic autoregressive moving average models. Technometrics 27, 375–384.CrossRefGoogle Scholar