Electronic Journal of Probability

Central Limit Theorems and Quadratic Variations in Terms of Spectral Density

Hermine Biermé, Aline Bonami, and José R. Leon

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Abstract

We give a new proof and provide new bounds for the speed of convergence in the Central Limit Theorem of Breuer Major on stationary Gaussian time series, which generalizes to particular triangular arrays. Our assumptions are given in terms of the spectral density of the time series. We then consider generalized quadratic variations of Gaussian fields with stationary increments under the assumption that their spectral density is asymptotically self-similar and prove Central Limit Theorems in this context.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 13, 362-395.

Dates
Accepted: 18 February 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820182

Digital Object Identifier
doi:10.1214/EJP.v16-862

Mathematical Reviews number (MathSciNet)
MR2774094

Zentralblatt MATH identifier
1238.60027

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 60G10: Stationary processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis 62M40: Random fields; image analysis 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Central limit theorem Gaussian stationary process spectral density periodogram quadratic variations fractional Brownian Motion

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Biermé, Hermine; Bonami, Aline; Leon, José R. Central Limit Theorems and Quadratic Variations in Terms of Spectral Density. Electron. J. Probab. 16 (2011), paper no. 13, 362--395. doi:10.1214/EJP.v16-862. https://projecteuclid.org/euclid.ejp/1464820182


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