New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

We extend Stein’s celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit symmetry of obtain error smooth test functions. also a continuous version in terms fourth moments. apply main results multivariate approximations Wishart matrices size n and degree d, where optimal convergence rate n3/d under only moment assumptions, degenerate U-statistics Poisson functionals, strengthen few bounds literature on distance.