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
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
