On quasi-infinite divisibility, limit theorems and signatures

In this thesis we investigate three main interconnected topics that lie in the interface between probability theory and mathematical statistics. They are quasi-infinite divisibility, limit theorems and signatures. Concerning quasi-infinite divisibility, we present numerous results on quasi-infinitely divisible (QID) distributions, processes and random measures. Lindner, Pan and Sato in \cite{LPS} show that QID distributions are dense in the space of all probability distributions with respect to weak convergence. We start our presentation with the results on QID random measures. In particular, we prove spectral representations, we build stochastic integral using them, and show necessary and sufficient conditions for their integrability. Then, we present several results on QID stochastic processes, including various spectral representations. The results presented in this chapter, namely Chapter \ref{Chaper-Quasi-infinite divisibility}, have been subsequently published in the article \cite{Pass-QID} and constitute the fundamental basis for the ongoing work \cite{Pass}, where it is proved that QID random measures are dense in the space of completely random measures under convergence in distribution. Concerning limit theorems, we present two main collection of results. In the first one, we derive some necessary and sufficient conditions for stationary ID multivariate random fields to be mixing. As an example we show that mixed moving average fields are mixing. In the second one, the main result is a central limit theorem for the multivariate Brownian semistationary (BSS) process. In particular, we study the joint asymptotic behaviour of its realised covariation using in-fill (\textit{i.e.}~high-frequency) asymptotics. For this central limit theorem we provide examples and feasible results, namely results that can be computed directly from data. These two collections of results will constitute the content of the works \cite{Pass-Mixing} and \cite{Pass-BSS}. Finally, we focus on the signature of the fractional Brownian motion (fBm) with Hurst parameter $H>\frac{1}{2}$. The signature of a $d$-dimensional fBm is a sequence of iterated Stratonovich integrals along the paths of the fBm. We provide a bound for the expected signature of the fBm and show the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value. Further, we present the general cubature method for the fBm with $H>\frac{1}{2}$ for small times. The cubature method is a numerical deterministic method based on signatures for the approximation of the expected solution of certain SDEs. These results will be published in the work \cite{Pass-Signature}.Open Acces