May 2023 Limit theorems for additive functionals of the fractional Brownian motion
Arturo Jaramillo, Ivan Nourdin, David Nualart, Giovanni Peccati
Author Affiliations +
Ann. Probab. 51(3): 1066-1111 (May 2023). DOI: 10.1214/22-AOP1612

Abstract

We investigate first and second order fluctuations of additive functionals of a fractional Brownian motion (fBm) of the form

(0.1){0tf(nH(Bsλ))ds;t0},

where B={Bt;t0} is a fBm with Hurst parameter H(0,1), f is a suitable test function and λR. We develop our study by distinguishing two regimes which exhibit different behaviors. When H(0,1/3), we show that a suitable renormalization of (), compensated by a multiple of the local time of B, converges toward a constant multiple of the derivative of the local time of B. In contrast, in the case H[1/3,1) we show that () converges toward an independent Brownian motion subordinated to the local time of B. Our results refine and complement those from (Ann. Appl. Probab. 31 (2021) 2143–2191), (Jeganathani (2006)), (Ann. Probab. 42 (2014) 168–203), (Electron. Commun. Probab. 74 (2013) 18) and solve at the same time the critical case H=1/3 which had remained open until now.

Funding Statement

Arturo Jaramillo Gil was supported by the FNR Grant R-AGR-3410-12-Z (MISSILe) at Luxembourg and by CONACYT Grant CB-2017-2018-A1-S-9764. Ivan Nourdin was supported by the FNR Grant APOGee (R-AGR-3585-10) at Luxembourg University. David Nualart was supported by the NSF Grant DMS 2054735. G. Peccati was supported by the FNR Grant FoRGES (R-AGR-3376-10) at Luxembourg University.

Acknowledgments

We would like to thank two anonymous referees for carefully reading the first version of this manuscript and making very valuable comments.

Citation

Download Citation

Arturo Jaramillo. Ivan Nourdin. David Nualart. Giovanni Peccati. "Limit theorems for additive functionals of the fractional Brownian motion." Ann. Probab. 51 (3) 1066 - 1111, May 2023. https://doi.org/10.1214/22-AOP1612

Information

Received: 1 August 2021; Revised: 1 August 2022; Published: May 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583063
zbMATH: 07690056
Digital Object Identifier: 10.1214/22-AOP1612

Subjects:
Primary: 60F05 , 60G22 , 60H07 , 60J55 , 62E17

Keywords: Additive functionals , Clark–Ocone formula , fractional Brownian motion , limit theorems , Local times

Rights: Copyright © 2023 Institute of Mathematical Statistics

JOURNAL ARTICLE
46 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.51 • No. 3 • May 2023
Back to Top