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

Abstract

This paper establishes an upper bound for the Kolmogorov distance between the maximum of a highdimensional 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.Let m * be an integer such that |ρ m * |Σ(θ m * ) = max 1≤m≤M |ρ m |Σ(θ m ). By assumption [A4], for each n ∈ N there is a number ϑ n ∈ G n such that |ϑ n − θ m * | ≤ υ n . Now noting (B.5), we havefor sufficiently large n. Let us denote by ⊖ the symmetric difference between two sets. Then we have

Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

“…For this purpose we need to verify conditions (3.7)-(3.17). Next, if a k-dimensional random vector ξ satisfy max 1≤i≤k ξ i p ≤ Ap r/2 for any p ∈ N with some constants A > 0 and r ∈ N, then Lemma A.7 and Proposition A.1 of [45] imply that ξ ℓ ∞ p ≤ A log r/2 (2k − 1 + e pr/2−1 ) for any p > 0 with pr ≥ 2. Using this fact, we can prove claim (b) in the same way as the proof of claim (a).…”

“…[10,16,70,71]; Chen [11] and Chen & Kato [12,13] have developed theories for U-statistics. Moreover, some authors have applied the CCK theory to statistical problems regarding high-frequency data; see Kato & Kurisu [40] and Koike [45]. Nevertheless, none of the above studies is applicable to our problem due to its non-ergodic nature.…”

High-dimensional econometrics and regularized GMM

“…CLT and bootstrap theorems to U -statistics and randomized incomplete U -statistics, respectively ( [42] focuses on the second order case). [81] studies Gaussian approximation to a high-dimensional vector of smooth Wiener functionals by combining the techniques developed in [45,49,50] and Malliavin calculus.…”

High-Dimensional Central Limit Theorems for Homogeneous Sums

“…On the one hand, however, the former are mainly concerned with non-degenerate U -statistics which are approximately linear statistics via Hoeffding decomposition (Chen & Kato [11] also handle degenerate U -statistics, but they focus on the randomized incomplete versions that are still approximately linear statistics). On the other hand, although the latter deal with essentially nonlinear statistics, they must be functionals of a (possibly infinitedimensional) Gaussian process, except for [35,Theorem 3.2] that is a version of our result with q j ≡ 2 (see Sect. 2.1.2 for more details).…”

“…Nevertheless, most studies focus on linear statistics (i.e., sums of random variables) and there are only a few articles concerned with nonlinear statistics. Two exceptions are U -statistics developed in [10][11][12]56] and Wiener functionals developed in [34,35]. On the one hand, however, the former are mainly concerned with non-degenerate U -statistics which are approximately linear statistics via Hoeffding decomposition (Chen & Kato [11] also handle degenerate U -statistics, but they focus on the randomized incomplete versions that are still approximately linear statistics).…”