Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral

Abstract

We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Ito integrals of which components all belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's method for normal approximation and method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergences for complex Wiener-Ito integrals, vector-valued Wiener-Ito integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals.