Central limit theorem for fourier transforms of stationary processes

Annals of Probability

Volumn 38, Issue 5, 2010, Pages 2009-2022

  Peligrad, Magda ^a   Wu, Wei Biao ^b  

^a UNIVERSITY OF CINCINNATI   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

Central limit theorem; Fourier transform; Martingale; Spectral analysis; Stationary process

Indexed keywords

EID: 77957245249     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOP530     Document Type: Article

References (30)

1. BARY, N. K. (1964). A Treatise on Trigonometric Series. Macmillan, New York.
2. BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
3. BRADLEY, R. C. (2007). Introduction to Strong Mixing Conditions 1, 2, 3. Kendrick Press, Heber City, UT.
4. BORODIN, A. N. and IBRAGIMOV, I. A. (1994). Limit theorems for functionals of random walks. Tr. Mat. Inst. Steklova 195 286.
5. BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
6. CARLESON, L. (1966). On convergence and growth of partial sumas of Fourier series. Acta Math. 116 135-157.
7. CHAMPENEY, D. C. (1989). A Handbook of Fourier Theorems. Cambridge Univ. Press, Cambridge.
8. DEDECKER, J. and RIO, E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 1-34.
9. DEDECKER, J. and MERLEVÈDE, F. (2002). Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 1044-1081.
10. FISHER, R. A. (1929). Tests of significance in harmonic analysis. Proc. Roy. Soc. Ser. A 125 54-59.
11. HALL, P. and HEYDE, C. C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. Academic Press, New York.
12. HUNT, R. A. and YOUNG, W. S. (1974). A weighted norm inequality for Fourier series. Bull. Amer. Math. Soc. 80 274-277.
13. KOLMOGOROV, A. (1923). Une série de Fourier-Lebesgue divergente presque partout. Fund. Math. 4 324-328.
14. LACEY, M. and TERWILLEGER, E. (2008). A Wiener-Wintner theorem for the Hilbert transform. Ark. Mat. 46 315-336.
15. LAHIRI, S. N. (2003). A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short-and long-range dependence. Ann. Statist. 31 613-641.
16. LIN, Z. and LIU, W. (2009). On maxima of periodograms of stationary processes. Ann. Statist. 37 2676-2695.
17. OLSHEN, R. A. (1967). Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Probab. 4 508-528.
18. ROOTŹN, H. (1976). Gordin's theorem and the periodogram. J. Appl. Probab. 13 365-370.
19. ROSENBLATT, M. (1981). Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrsch. Verw. Gebiete 55 123-132.
20. ROSENBLATT, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston, MA.
21. SCHUSTER, A. (1898). On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena. Terrestrial Magnetism and Atmospheric Electricity 3 13-41.
22. TERRIN, N. and HURVICH, C. M. (1994). An asymptotic Wiener-Itô representation for the low frequency ordinates of the periodogram of a long memory time series. Stochastic Process. Appl. 54 297-307.
23. WALKER, A. M. (1965). Some asymptotic results for the periodogram of a stationary time series. J. Aust. Math. Soc. 5 107-128.
24. WALKER, A. M. (2000). Some results concerning the asymptotic distribution of sample Fourier transforms and periodograms for a discrete-time stationary process with a continuous spectrum. J. Time Ser. Anal. 21 95-109.
25. WOODROOFE, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic Process. Appl. 41 33-44.
26. WU, W. B. and WOODROOFE, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748-755.
27. WU, W. B. (2005). Fourier transforms of stationary processes. Proc. Amer. Math. Soc. 133 285-293.
28. YAJIMA, Y. (1989). A central limit theorem of Fourier transforms of strongly dependent stationary processes. J. Time Ser. Anal. 10 375-383.
29. WIENER, N. and WINTNER, A. (1941). On the ergodic dynamics of almost periodic systems. Amer. J. Math. 63 794-824.
30. ZYGMUND, A. (2002). Trigonometric Series. Cambridge Univ. Press, Cambridge.