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. <script type=text/javascript src=//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML>
Highlights
The quadratic variation of a stochastic process X plays an important role in different applications
To obtain the generality we study sequences of general n-dimensional Gaussian vectors Y n = (Y1(n), . . . , Yn(n)), where each component Yk(n) may depend on n, and we study the asymptotic behaviour of the vector Y n or its quadratic variation defined as the limit n lim n→∞
We show that, rather surprisingly, the quadratic variation always converges to the energy of the process whether or not the energy is finite provided that 2-planar variation vanishes
Summary
The quadratic variation of a stochastic process X plays an important role in different applications. The central limit theorem for a more general sequence f (Xt), where Xt is a stationary Gaussian process, was proved in [12] Later this result was refined in [48] to obtain functional convergence to the fractional Brownian motion or to the Rosenblatt process. We begin by providing necessary and sufficient conditions under which appropriately scaled quadratic variation converges to a Gaussian limit To obtain this result we apply a powerful fourth moment theorem proved by Nualart and Peccati [38] which, thanks to the recent results by Sottinen and the current author [47], can essentially be applied always. In this paper we give necessary and sufficient conditions for the convergence of quadratic variations of general Gaussian vectors which can be used to reproduce and generalise existing results.
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