Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences

Abstract

The quadratic variation of Gaussian processes plays an 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 convergence of quadratic sums of general Gaussian sequences. We provide necessary and sufficient conditions for different types of convergence including convergence in probability, almost sure convergence, $L^{p}$-convergence as well as weak convergence. We use a practical and simple approach which simplifies the existing methodology considerably. As an application, we show how convergence of the quadratic variation of a given process can be obtained by an appropriate choice of the underlying sequence.