Asymptotic behavior of local times related statistics for fractional Brownian motion

TL;DR

This paper studies the asymptotic behavior of local time-related statistics for fractional Brownian motion observed at high frequency, revealing convergence to local times and deriving limit theorems for quadratic variation.

Contribution

It extends classical local time statistics analysis from diffusion processes to fractional Brownian motion, establishing convergence results and limit theorems.

Findings

Statistics converge to local times up to a constant

Limit theorems for quadratic variation of fractional Brownian motion

Provides asymptotic behavior insights for high-frequency data

Abstract

We consider high frequency observations from a fractional Brownian motion. Inspired by the work of Jean Jacod in a diffusion setting, we investigate the asymptotic behavior of various classical statistics related to the local times of the process. We show that as in the diffusion case, these statistics indeed converge to some local times up to a constant factor. As a corollary, we provide limit theorems for the quadratic variation of the absolute value of a fractional Brownian motion.