Fourier transforms of stationary processes

Author:
Wei Biao Wu

Journal:
Proc. Amer. Math. Soc. **133** (2005), 285-293

MSC (2000):
Primary 60F05, 60F17; Secondary 60G35

DOI:
https://doi.org/10.1090/S0002-9939-04-07528-8

Published electronically:
May 20, 2004

MathSciNet review:
2086221

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the asymptotic behavior of Fourier transforms of stationary and ergodic sequences. Under sufficiently mild conditions, central limit theorems are established for almost all frequencies as well as for a given frequency. Applications to the widely used linear processes and iterated random functions are discussed. Our results shed new light on the foundation of spectral analysis in that the asymptotic distribution of the periodogram, the fundamental quantity in the frequency-domain analysis, is obtained.

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Additional Information

**Wei Biao Wu**

Affiliation:
Department of Statistics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Email:
wbwu@galton.uchicago.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07528-8

Keywords:
Spectral analysis,
linear process,
martingale central limit theorem,
periodogram,
Fourier transformation,
nonlinear time series

Received by editor(s):
March 24, 2003

Received by editor(s) in revised form:
June 27, 2003, and September 18, 2003

Published electronically:
May 20, 2004

Communicated by:
Richard C. Bradley

Article copyright:
© Copyright 2004
American Mathematical Society