Limit theorems for high-frequency data: from semimartingales to ambit fields

Understanding the asymptotic behavior of statistics based on high frequency observations has gained considerable attention in the recent years, notably due to the ever-growing availability of datas. We think of the numerous applications in economics and econometrics, among others. This thesis explores the limit theory of realized quadratic variation and related functionals for two important classes of processes, namely semimartingales and ambit fields. Chapter I is an introductory chapter to the main mathematical concepts encountered in the thesis. We preface the classes of semimartingales and of ambit fields and introduce the commom problematic to all the following chapters, that is the establishment of the asymptotic theory for functionals of increments of the processes of interest. We finally present the different methodologies to answer this problematic in the various chapters of the sequel. Chapter II contains the paper [44]: ”Limit theorems for general functionals of Brownian local times”, in collaboration with Simon Campese and Mark Podolskij. Electronic Journal of Probability, 29:1–18, 2024. We prove a stable central limit theorem for a class of integrated functionals of increments of the local time of a Brownian motion. This result generalizes a number of prior works in the unified framework of semimartingales’ limit theory. Chapter III contains the preprint: ”Limit theorems for asynchronously observed bivariate pure jump semimartingales”, in collaboration with Mark Podolskij, 2024. In this chapter we prove a non-central limit theorem for the Hayashi-Yoshida estimator of the quadratic covariation process of an asynchronously observed stable process. This result is one of the first to establish the asymptotic theory for nonsynchronous high-frequency statistics of pure jump processes. Chapter IV contains the preprint: ”Limit theorems for two dimensional ambit fields observed along curves”, in collaboration with Mikko S. Pakkanen, Mark Podolskij and Bezirgen Veliyev, work in progress. The main result is a stable central limit theorem for the power variation of increments of a two-dimensional ambit field, observed with high frequency along some curve embedded in the field. This result is a direct extension of the univariate case. Finally, the appendix contains technical results that complement the various concepts covered in the introduction.