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2022 Limit theorems for Toeplitz-type quadratic functionals of stationary processes and applications
Mamikon S. Ginovyan, Murad S. Taqqu
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Probab. Surveys 19: 1-64 (2022). DOI: 10.1214/21-PS3

Abstract

This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.

Funding Statement

Murad S. Taqqu was supported in part by a Simons Foundation grant #569118 at Boston University.

Acknowledgments

The authors would like to thank the anonymous referees for their careful review of the manuscript and valuable remarks and comments.

Citation

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Mamikon S. Ginovyan. Murad S. Taqqu. "Limit theorems for Toeplitz-type quadratic functionals of stationary processes and applications." Probab. Surveys 19 1 - 64, 2022. https://doi.org/10.1214/21-PS3

Information

Received: 1 February 2021; Published: 2022
First available in Project Euclid: 17 January 2022

Digital Object Identifier: 10.1214/21-PS3

Subjects:
Primary: 60F05 , 60G10 , 60G15
Secondary: 62F12 , 62G05

Keywords: Brownian motion , central and non-central limit theorems , parametric and nonparametric estimation , Spectral density , Stationary processes , Toeplitz type quadratic functional

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Vol.19 • 2022
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