Rates for the CLT Via New Ideal Metrics

Abstract

Let $(B, \| \|)$ be a separable Banach space and $\mathscr{X} := \mathscr{X}(B)$ the vector space of all random variables defined on a probability space $(\Omega, \mathscr{A}, P)$ and taking values in $B$. It is shown that new ideal metrics for $\mathscr{X}$ may be used to obtain refined rates of convergence of normalized sums to a stable limit law. The rates hold uniformly in $n$ and are expressed in terms of a variety of uniform metrics on $\mathscr{X}$. In the Banach space setting the rates hold with respect to the total variation metric and in the Euclidean space setting the rates hold with respect to uniform metrics between density and characteristic functions. The main result provides a sharp order estimate of the rate of convergence in local limit theorems with respect to the uniform distance between densities. The method is based on the theory of probability metrics, especially those of convolution type.