Convergence of some random functionals of discretized semimartingales

TL;DR

This paper investigates the asymptotic behavior of sums of functions of discretized semimartingale increments, providing convergence results and central limit theorems relevant for statistical applications.

Contribution

It extends existing results by analyzing sums where the function depends on current and past information of the semimartingale, not just the increments.

Findings

Convergence in probability of sums of functionals of semimartingale increments.

Central limit theorems for these sums.

Extension of known results to more general functional dependencies.

Abstract

In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also on the past of the semimartingale before this time. We study the convergence in probability of two types of such sums, and we also give associated central limit theorems. This extends known results when the summands are a function depending only on the increments, and this is motivated mainly by statistical applications.