## Bernoulli

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

### Quenched limit theorems for Fourier transforms and periodogram

#### 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
Revised: April 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620850

Digital Object Identifier
doi:10.3150/14-BEJ658

Mathematical Reviews number (MathSciNet)
MR3449783

Zentralblatt MATH identifier
1336.60038

#### 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

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