Bernoulli

  • Bernoulli
  • Volume 22, Number 1 (2016), 275-301.

Quenched limit theorems for Fourier transforms and periodogram

David Barrera and Magda Peligrad

Full-text: Open access

Abstract

In this paper, we study the quenched central limit theorem for the discrete Fourier transform. We show that the Fourier transform of a stationary ergodic process, suitable centered and normalized, satisfies the quenched CLT conditioned by the past sigma algebra. For functions of Markov chains with stationary transitions, this means that the CLT holds with respect to the law of the chain started at a point for almost all starting points. It is necessary to emphasize that no assumption of irreducibility with respect to a measure or other regularity conditions are imposed for this result. We also discuss necessary and sufficient conditions for the validity of quenched CLT without centering. The results are highly relevant for the study of the periodogram of a Markov process with stationary transitions which does not start from equilibrium. The proofs are based of a nice blend of harmonic analysis, theory of stationary processes, martingale approximation and ergodic theory.

Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 275-301.

Dates
Received: January 2014
Revised: April 2014
First available in Project Euclid: 30 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1443620850

Digital Object Identifier
doi:10.3150/14-BEJ658

Mathematical Reviews number (MathSciNet)
MR3449783

Zentralblatt MATH identifier
1336.60038

Keywords
central limit theorem discrete Fourier transform martingale approximation periodogram spectral analysis

Citation

Barrera, David; Peligrad, Magda. Quenched limit theorems for Fourier transforms and periodogram. Bernoulli 22 (2016), no. 1, 275--301. doi:10.3150/14-BEJ658. https://projecteuclid.org/euclid.bj/1443620850


Export citation

References

  • [1] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
  • [2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley.
  • [3] Bradley, R.C. (2007). Introduction to Strong Mixing Conditions. Vol. 1. Heber City, UT: Kendrick Press.
  • [4] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [5] Carleson, L. (1966). On convergence and growth of partial sums of Fourier series. Acta Math. 116 135–157.
  • [6] Cohen, G. and Conze, J.-P. (2013). The CLT for rotated ergodic sums and related processes. Discrete Contin. Dyn. Syst. 33 3981–4002.
  • [7] Cuny, C. and Merlevède, F. (2014). On martingale approximations and the quenched weak invariance principle. Ann. Probab. 42 760–793.
  • [8] Cuny, C., Merlevède, F. and Peligrad, M. (2013). Law of the iterated logarithm for the periodogram. Stochastic Process. Appl. 123 4065–4089.
  • [9] Cuny, C. and Peligrad, M. (2012). Central limit theorem started at a point for stationary processes and additive functionals of reversible Markov chains. J. Theoret. Probab. 25 171–188.
  • [10] Cuny, C. and Volný, D. (2013). A quenched invariance principle for stationary processes. ALEA Lat. Am. J. Probab. Math. Stat. 10 107–115.
  • [11] Dedecker, J., Gouëzel, S. and Merlevède, F. (2010). Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 46 796–821.
  • [12] Dedecker, J., Merlevède, F. and Peligrad, M. (2014). A quenched weak invariance principle. Ann. Inst. Henri Poincaré Probab. Stat. 50 872–898.
  • [13] Derriennic, Y. and Lin, M. (2001). The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Related Fields 119 508–528.
  • [14] de la Peña, V.H. and Giné, E. (1999). Decoupling: From Dependence to Independence, Randomly Stopped Processes. $U$-Statistics and Processes. Martingales and Beyond. Probability and Its Applications (New York). New York: Springer.
  • [15] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76.
  • [16] Dunford, N. and Schwartz, J.T. (1988). Linear Operators: General Theory. Part I. Wiley Classics Library. New York: Wiley. With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original.
  • [17] Eisner, T., Farkas, B., Haase, M. and Nagel, R. (2012). Operator Theoretic Aspects of Ergodic Theory. Berlin: Springer.
  • [18] Gänssler, P. and Häusler, E. (1979). Remarks on the functional central limit theorem for martingales. Z. Wahrsch. Verw. Gebiete 50 237–243.
  • [19] Gordin, M.I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739–741.
  • [20] Gordin, M.I. and Lifšic, B.A. (1981). A remark about a Markov process with normal transition operator. In Third Vilnius Conf. Proba. Stat., Akad. Nauk Litovsk, Vol. 1 147–148 (in Russian), Vilnius.
  • [21] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. Probability and Mathematical Statistics. New York: Academic Press.
  • [22] Herrndorf, N. (1983). Stationary strongly mixing sequences not satisfying the central limit theorem. Ann. Probab. 11 809–813.
  • [23] Hunt, R.A. and Young, W.S. (1974). A weighted norm inequality for Fourier series. Bull. Amer. Math. Soc. 80 274–277.
  • [24] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
  • [25] Peligrad, M. (2013). Quenched Invariance Principles Via Martingale Approximation. Fields Institute Communications. Springer. To appear. Available at http://arxiv.org/pdf/1304.4580.pdf.
  • [26] Peligrad, M. and Wu, W.B. (2010). Central limit theorem for Fourier transforms of stationary processes. Ann. Probab. 38 2009–2022.
  • [27] Rassoul-Agha, F. and Seppäläinen, T. (2007). Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab. 35 1–31.
  • [28] Rassoul-Agha, F. and Seppäläinen, T. (2008). An almost sure invariance principle for additive functionals of Markov chains. Statist. Probab. Lett. 78 854–860.
  • [29] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Berlin: Springer.
  • [30] Rootzén, H. (1976). Gordin’s theorem and the periodogram. J. Appl. Probab. 13 365–370.
  • [31] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
  • [32] Rosenblatt, M. (1961). Independence and dependence. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 431–443. Berkeley, CA: Univ. California Press.
  • [33] Rosenblatt, M. (1981). Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrsch. Verw. Gebiete 55 123–132.
  • [34] Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Boston, MA: Birkhäuser.
  • [35] 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.
  • [36] 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.
  • [37] Volný, D. and Woodroofe, M. (2010). An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process. In Dependence in Probability, Analysis and Number Theory 317–322. Heber City, UT: Kendrick Press.
  • [38] Walker, A.M. (1965). Some asymptotic results for the periodogram of a stationary time series. J. Aust. Math. Soc. 5 107–128.
  • [39] Wiener, N. and Wintner, A. (1941). On the ergodic dynamics of almost periodic systems. Amer. J. Math. 63 794–824.
  • [40] Woodroofe, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic Process. Appl. 41 33–44.
  • [41] Wu, W.B. (2005). Fourier transforms of stationary processes. Proc. Amer. Math. Soc. 133 285–293 (electronic).
  • [42] Wu, W.B. and Woodroofe, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748–755.