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Barrera, David; Peligrad, Magda

Quenched limit theorems for Fourier transforms and periodogram. (English) Zbl 1336.60038

Bernoulli 22, No. 1, 275-301 (2016).
Summary: 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, suitably 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 the 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 on a nice blend of harmonic analysis, theory of stationary processes, martingale approximation and ergodic theory.

Cited in 2 Documents

MSC:

60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
60G10 Stationary stochastic processes
60G42 Martingales with discrete parameter
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

quenched central limit theorem; discrete Fourier transform; Markov chains; periodogram; stationary processes; martingale approximation; spectral analysis; ergodic theory
PDFBibTeX XMLCite
\textit{D. Barrera} and \textit{M. Peligrad}, Bernoulli 22, No. 1, 275--301 (2016; Zbl 1336.60038)
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References:

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