Limit Theorems for Additive Functionals of the Fractional Brownian Motion

TL;DR

This paper establishes limit theorems for additive functionals of fractional Brownian motion, revealing different behaviors in sub- and super-critical regimes and solving the open case at H=1/3.

Contribution

It provides a comprehensive analysis of the fluctuations of additive functionals of fBm across different Hurst parameter regimes, including the critical case H=1/3.

Findings

For H in (0,1/3), the renormalized functional converges to a derivative of local time.

For H in [1/3,1), the functional converges to a Brownian motion subordinated to local time.

The critical case H=1/3 is explicitly solved, completing the understanding of the behavior.

Abstract

We investigate first and second order fluctuations of additive functionals of a fractional Brownian motion (fBm) of the form \begin{align}\label{eq:abstractmain} Z_n=\left\{\int_{0}^{t}f(n^{H}(B_{s}-\lambda))ds\ ; t\geq 0 \right\} \end{align} where $B=\{B_{t}; t \geq 0\}$ is a fBm with Hurst parameter $H\in (0,1)$, $f$ is a suitable test function and $\lambda\in \mathbb{R}$. We develop our study by distinguishing two regimes which exhibit different behaviors. When $H\in(0,1/3)$, we show that a suitable renormalization of $Z_n$, compensated by a multiple of the local time of $B$, converges towards a constant multiple of the derivative of the local time of $B$. In contrast, in the case $H\in[1/3,1)$ we show that $Z_n$ converges towards an independent Brownian motion subordinated to the local time of $B$. Our results refine and complement those from the current literature and solve at the same time the critical case $H=1/3$, which had remained open until now.