Kolmogorov distance for the central limit theorems of the Wiener chaos expansion and applications

This paper concerns the rate of convergence for the central limit theorems of the chaos expansion of functionals of Gaussian process. The aim of the present work is to derive upper bounds of the Kolmogorov distance for the rate of convergence. We apply our results to find the upper bound of the Kolmogorov distance in the quantitative Breuer–Major theorems (Nourdin et al., 2011), and prove that the upper bound in our results is more efficient than that in the quantitative Breuer–Major theorems. Also we obtain the explicit upper bound of the Kolmogorov distance for central limit theorems of sojourn times.