  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.
