Contributions to the asymptotic study of Hermite driven processes

TL;DR

This thesis explores the asymptotic properties of Hermite-driven processes, including limit theorems, quadratic variations, and statistical inference, advancing understanding of their long-memory and non-Gaussian behaviors.

Contribution

It provides new limit theorems and statistical methods specifically for Hermite processes and their quadratic functionals, extending prior work on non-Gaussian long-memory processes.

Findings

Non-central limit theorem for quadratic functionals of Hermite processes

Asymptotic behavior of quadratic variations in multiparameter Hermite fields

Fisher information related results for Hermite-driven models

Abstract

This thesis consists of two parts. Part I is an introduction to Hermite processes, Hermite random fields, Fisher information and to the papers constituting the thesis. More precisely, in Section 1 we introduce Hermite processes in a nutshell, as well as some of its basic properties. It is the necessary background for the articles [a] and [c]. In Section 2 we consider briefly the multiparameter Hermite random fields and we study some less elementary facts which are used in the article [b]. In section 3, we recall some terminology about Fisher information related to the article [d]. Finally, our articles [a] to [d] are summarised in Section 4. Part II consists of the articles themselves: [a] T.T. Diu Tran (2017): Non-central limit theorem for quadratic functionals of Hermite-driven long memory moving average processes. \textit{Stochastic and Dynamics}, \textbf{18}, no. 4. [b] T.T. Diu Tran (2016): Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields. Under revision for \textit{Probability and Mathematical Statistics}. [c] I. Nourdin, T.T. Diu Tran (2017): Statistical inference for Vasicek-type model driven by Hermite processes. Submitted to \textit{Stochastic Process and their Applications}. [d] T.T. Diu Tran (2017+): Fisher information and multivariate Fouth Moment Theorem. Main results have already been obtained. It should be submitted soon.