The Annals of Statistics

Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data

Yuta Koike

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This paper establishes an upper bound for the Kolmogorov distance between the maximum of a high-dimensional vector of smooth Wiener functionals and the maximum of a Gaussian random vector. As a special case, we show that the maximum of multiple Wiener–Itô integrals with common orders is well approximated by its Gaussian analog in terms of the Kolmogorov distance if their covariance matrices are close to each other and the maximum of the fourth cumulants of the multiple Wiener–Itô integrals is close to zero. This may be viewed as a new kind of fourth moment phenomenon, which has attracted considerable attention in the recent studies of probability. This type of Gaussian approximation result has many potential applications to statistics. To illustrate this point, we present two statistical applications in high-frequency financial econometrics: One is the hypothesis testing problem for the absence of lead-lag effects and the other is the construction of uniform confidence bands for spot volatility.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1663-1687.

Received: September 2017
Revised: March 2018
First available in Project Euclid: 13 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus 62E17: Approximations to distributions (nonasymptotic) 62G20: Asymptotic properties

Bootstrap fourth moment phenomenon Malliavin calculus maximum Stein’s method uniform confidence bands


Koike, Yuta. Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data. Ann. Statist. 47 (2019), no. 3, 1663--1687. doi:10.1214/18-AOS1731.

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Supplemental materials

  • Supplement to “Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data”. This supplement file contains the technical materials of the paper and consists of two appendices. Appendix A collects the preliminary definitions and results used in Appendix B, which contains proofs of all the results presented in the main text of the paper.