Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes

TL;DR

This paper proves weak convergence of normalized Toeplitz quadratic functionals of continuous-time Gaussian stationary processes with long-range dependence, establishing both central and non-central limit theorems.

Contribution

It introduces new weak convergence results for Toeplitz quadratic functionals in continuous time, covering both central and non-central limit theorems.

Findings

Weak convergence in C[0,1] for normalized functionals

Establishment of central limit theorems

Derivation of non-central limit theorems

Abstract

\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained.