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

TL;DR

This paper develops a Gaussian approximation for the maximum of Wiener functionals, providing bounds on their distributional distance, with applications to high-frequency financial econometrics such as hypothesis testing and confidence band construction.

Contribution

It introduces a new Gaussian approximation method for maxima of Wiener functionals with explicit bounds, applicable to high-frequency data analysis.

Findings

Gaussian bounds for maxima of Wiener functionals established

Approximation accuracy depends on covariance closeness and fourth cumulant

Applications include hypothesis testing and confidence bands in finance

Abstract

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-It\^o 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-It\^o 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.