August 2021 Glivenko–Cantelli theorems for integrated functionals of stochastic processes
Jia Li, Congshan Zhang, Yunxiao Liu
Author Affiliations +
Ann. Appl. Probab. 31(4): 1914-1943 (August 2021). DOI: 10.1214/20-AAP1637

Abstract

We prove a Glivenko–Cantelli theorem for integrated functionals of latent continuous-time stochastic processes. Based on a bracketing condition via random brackets, the theorem establishes the uniform convergence of a sequence of empirical occupation measures towards the occupation measure induced by underlying processes over large classes of test functions, including indicator functions, bounded monotone functions, Lipschitz-in-parameter functions, and Hölder classes as special cases. The general Glivenko–Cantelli theorem is then applied in more concrete high-frequency statistical settings to establish uniform convergence results for general integrated functionals of the volatility of efficient price and local moments of microstructure noise.

Acknowledgements

We would like to thank Tim Bollerslev and George Tauchen for helpful discussions.

Citation

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Jia Li. Congshan Zhang. Yunxiao Liu. "Glivenko–Cantelli theorems for integrated functionals of stochastic processes." Ann. Appl. Probab. 31 (4) 1914 - 1943, August 2021. https://doi.org/10.1214/20-AAP1637

Information

Received: 1 October 2019; Revised: 1 July 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

MathSciNet: MR4312850
zbMATH: 1476.60069
Digital Object Identifier: 10.1214/20-AAP1637

Subjects:
Primary: 60F17 , 60G44 , 60G57 , 62G05

Keywords: Glivenko–Cantelli , high-frequency data , microstructure noise , occupation measure , spot volatility

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.31 • No. 4 • August 2021
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