A Survey on Limit Theorems for Toeplitz Type Quadratic Functionals of Stationary Processes and Applications

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 L\'evy-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, Fej\'er-type singular integrals, and L\'evy-It\^o-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.