October 2021 Approximation of fractional local times: Zero energy and derivatives
Arturo Jaramillo, Ivan Nourdin, Giovanni Peccati
Author Affiliations +
Ann. Appl. Probab. 31(5): 2143-2191 (October 2021). DOI: 10.1214/20-AAP1643


We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) X with Hurst parameter H(0,1), and derive conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order (possibly uniform) limit theorem. We devote specific emphasis to the “zero energy” case, corresponding to a kernel whose integral on the real line equals zero. Our asymptotic results are associated with explicit rates of convergence, and are expressed either in terms of the local time of X or of its derivatives: in particular, the full force of our finding applies to the “rough range” 0<H<1/3, on which the previous literature has been mostly silent. The use of the derivatives of local times for studying the fluctuations of high-frequency observations of a fBm is new, and is the main technological breakthrough of the present paper. Our results are based on the use of Malliavin calculus and Fourier analysis, and extend and complete several findings in the literature, for example, by Jeganathan (Ann. Probab. 32 (2004) 1771–1795; (2006); (2008)) and Podolskij and Rosenbaum (J. Financ. Econom. 16 (2018) 588–598).

Funding Statement

A. Jaramillo is supported by the FNR grant R-AGR-3410-12-Z (MISSILe) at Luxembourg and Singapore Universities. G. Peccati is supported by the FNR grant R-AGR-3376-10 (FoRGES) at Luxembourg University.


We thank Mark Podolskij for a number of illuminating discussions.


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Arturo Jaramillo. Ivan Nourdin. Giovanni Peccati. "Approximation of fractional local times: Zero energy and derivatives." Ann. Appl. Probab. 31 (5) 2143 - 2191, October 2021. https://doi.org/10.1214/20-AAP1643


Received: 1 April 2019; Revised: 1 March 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

MathSciNet: MR4332693
zbMATH: 1476.60068
Digital Object Identifier: 10.1214/20-AAP1643

Primary: 60F17 , 60G22 , 60H07 , 60J55

Keywords: derivatives of the local time , fractional Brownian motion , Functional limit theorems , high-frequency observations , Malliavin calculus

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 5 • October 2021
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