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Pre-averaging fractional processes contaminated by noise, with an application to turbulence

Part of: Nonparametric inference Turbulence Stochastic processes Inference from stochastic processes Basic methods in fluid mechanics

Published online by Cambridge University Press:  21 November 2025

David Chen ,
Yu Cheng ,
Carsten H. Chong ,
Pierre Gentine ,
Wangdong Jia ,
Bryce J. Monier  and
Shiyang Shen
David Chen*
Affiliation:
University of Chicago
Yu Cheng*
Affiliation:
Harvard University
Carsten H. Chong*
Affiliation:
The Hong Kong University of Science and Technology
Pierre Gentine*
Affiliation:
Columbia University
Wangdong Jia*
Affiliation:
University of California, Berkeley
Bryce J. Monier*
Affiliation:
Columbia University
Shiyang Shen*
Affiliation:
California Institute of Technology
*
*Postal address: Department of Statistics, University of Chicago, 5747 South Ellis Avenue, Chicago, IL 60637, USA. Email: davchen@uchicago.edu
**Postal address: Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, MA 02138, USA. Email: yucheng1@fas.harvard.edu
***Postal address: Department of Information Systems, Business Statistics and Operations Management, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: carstenchong@ust.hk
****Postal address: Department of Earth and Environmental Engineering and Earth Institute, Columbia University, 500 West 120th Street, New York, NY 10027, USA. Email: pg2328@columbia.edu
*****Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, 4141 Etcheverry Hall, Berkeley, CA 94720, USA. Email: wangdong_jia@berkeley.edu
******Postal address: Columbia College, Columbia University, 1130 Amsterdam Avenue, New York, NY 10027, USA. Email: bjm2190@columbia.edu
*******Postal address: The Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA. Email: sshen2@caltech.edu
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Abstract

We consider the problem of estimating fractional processes based on noisy high-frequency data. Generalizing the idea of pre-averaging to a fractional setting, we exhibit a sequence of consistent estimators for the unknown parameters of interest by proving a law of large numbers for associated variation functionals. In contrast to the semimartingale setting, the optimal window size for pre-averaging depends on the unknown roughness parameter of the underlying process. We evaluate the performance of our estimators in a simulation study and use them to empirically verify Kolmogorov’s $2/3$-law in turbulence data contaminated by instrument noise.

Keywords

Fractional Brownian motionhigh-frequency dataHurst parameter, Kolmogorov’s 2/3-lawlaw of large numbersnoisy observations

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 62M09: Non-Markovian processes 76M35: Stochastic analysis
Secondary: 62G05: Estimation 76F55: Statistical turbulence modeling

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 4 , December 2025 , pp. 1280 - 1300
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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