  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.

Probability