Sufficient and Necessary Conditions for Limit Theorems for Quadratic Variations of Gaussian Sequences

TL;DR

This paper establishes comprehensive conditions for various types of convergence of quadratic variations in Gaussian processes, enhancing theoretical understanding and simplifying analysis methods in stochastic processes.

Contribution

It provides the first unified framework of necessary and sufficient conditions for multiple convergence types in Gaussian quadratic variations, including a simplified approach.

Findings

Conditions for convergence in probability, almost sure, $L^p$, and law.

Application to Gaussian vectors for different quadratic variation cases.

A simplified methodology for analyzing convergence.

Abstract

Quadratic variations of Gaussian processes play important role in both stochastic analysis and in applications such as estimation of model parameters, and for this reason the topic has been extensively studied in the literature. In this article we study the problem for general Gaussian processes and we provide sufficient and necessary conditions for different types of convergence which include convergence in probability, almost sure convergence, $L^p$-convergence as well as convergence in law. Furthermore, we study general Gaussian vectors from which different interesting cases including first or second order quadratic variations can be studied by appropriate choice of the underlying vector. Finally, we provide a practical and simple approach to attack the problem which simplifies the existing methodology considerably.