GAUSSIAN APPROXIMATION OF MAXIMA OF WIENER FUNCTIONALS AND ITS APPLICATION TO HIGH-FREQUENCY DATA

• 成果类型: Article
• 署名作者: Koike, Yuta
• 署名单位: University of Tokyo; Japan Science & Technology Agency (JST)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1731
• 发表日期: 2019
• 页码: 1663-1687
• 关键词: CENTRAL LIMIT-THEOREMS; spot volatility; bootstrap; jumps; statistics; Invariance; expansion; suprema; noise; sums

摘要

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-Ito 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-Ito 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.