Probability Surveys

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

Lauri Viitasaari

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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.

Article information

Source
Probab. Surveys, Volume 16 (2019), 62-98.

Dates
Received: October 2015
First available in Project Euclid: 31 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.ps/1559289658

Digital Object Identifier
doi:10.1214/15-PS267

Mathematical Reviews number (MathSciNet)
MR3960291

Zentralblatt MATH identifier
07064382

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F25: $L^p$-limit theorems

Keywords
Quadratic variations Gaussian sequences Gaussian processes convergence in probability strong convergence convergence in $L^{p}$ central limit theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Viitasaari, Lauri. Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences. Probab. Surveys 16 (2019), 62--98. doi:10.1214/15-PS267. https://projecteuclid.org/euclid.ps/1559289658


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