Open AccessNecessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences
Author(s) -
Lauri Viitasaari
Publication year - 2019
Publication title -
probability surveys
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.49
H-Index - 43
ISSN - 1549-5787
DOI - 10.1214/15-ps267
Subject(s) - mathematics , quadratic variation , convergence (economics) , gaussian , quadratic equation , limit (mathematics) , sequence (biology) , weak convergence , convergence tests , convergence of random variables , modes of convergence (annotated index) , compact convergence , gaussian process , normal convergence , mathematical optimization , simple (philosophy) , rate of convergence , computer science , mathematical analysis , pure mathematics , key (lock) , random variable , statistics , brownian motion , geometry , quantum mechanics , physics , philosophy , computer security , isolated point , asset (computer security) , economic growth , topological vector space , biology , genetics , epistemology , topological space , economics
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.