On maxima of periodograms of stationary processes

Annals of Statistics

Volumn 37, Issue 5 B, 2009, Pages 2676-2695

  Lin, Zhengyan ^a   Liu, Weidong ^a  

^a ZHEJIANG UNIVERSITY   (China)

Author keywords

M dependent approximation; Periodogram; Stationary process

Indexed keywords

EID: 69049105438     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-AOS590     Document Type: Article

References (21)

1. An, H.Z., Chen, Z.G. and Hannan, E. J. (1983). The maximum of the periodogram. J. Multivariate Anal. 13 383-400. MR0716931
2. Brockwell, P. J. and Davis, R. A. (1998). Time Series: Theory and Methods, 2nd ed. Springer, New York. MR0868859
3. Chiu, S. T. (1989). Detecting periodic components in a white Gaussian time series. J. Roy. Statist. Soc. Ser. B 51 249-259. MR1007457
4. Davis, R.A. and Mikosch, T. (1999). The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. 27 522-536. MR1681157
5. Einmahl, U. (1989). Extensions of results of Komlos, Major, and Tusnady to the multivariate case. J. Multivariate Anal. 28 20-68. MR0996984
6. Fisher, R. A. (1929). Tests of significance in harmonic analysis. Proc. Roy. Statist. Soc. Ser. A 125 54-59.
7. Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Probab. 3 100-118. MR0380971
8. Lewis, T. and Fieller, N. R. J. (1979). A recursive algorithm for null distributions for outliers: I. Gamma samples. Technometrics 21 371-376. MR0538441
9. Hannan, E. J. (1961). Testing for a jump in the spectral function. J. Roy. Statist. Soc. Ser. B 23 394-404. MR0137271
10. Hannan, E. J. (1970). Multiple Time Series 463-475. Wiley, New York. MR0279952
11. Hsing, T. and Wu, W. B. (2004). On weighted U-statistics for stationary processes. Ann. Probab. 32 1600-1631. MR2060311
12. Lin, Z.Y. and Liu, W. D. (2008). Supplementary material for "On maxima of periodograms of stationary processes." Available at http://www.math.zju.edu.cn/stat/linliu.pdf; http://xxx. arxiv.org/abs/0801.1357.
13. Nagaev, S. V. (1979). Large deviations of independent random variables. Ann. Probab. 7 745-789. MR0542129
14. Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic Press, London.
15. Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773-1801. MR2351105
16. Shimshoni, M. (1971). On Fisher's test of significance in harmonic analysis. Geophys. J. R. Astronom. Soc. 23 373-377.
17. Walker, A. M. (1965). Some asymptotic results for the periodogram of a stationary time series. J. Aust. Math. Soc. 5 107-128. MR0177457
18. Woodroofe, M. and van Ness, J. W. (1967). The maximum deviation of sample spectral densities. Ann. Math. Statist. 38 1558-1569. MR0216717
19. Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150-14154. MR2172215
20. Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294-2320. MR2353389
21. Wu, W.B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425-436. MR2052582