Estimates of transport distance in the central limit theorem

Abstract

Let $X_1,\ldots,X_n$ be $d$-dimensional independent random vectors bounded with probability one. For simplicity, we assume that they have zero mean values: \begin{equation} \mathbf{P}\{\|X_{j}\|\le\tau\}=1,\quad\mathbf{E}\,X_{j}=0,\quad j=1,\ldots, n.\nonumber \end{equation} We study the distribution behavior of the sum $S=X_{1}+\cdots+X_{n}$ as a function of the bounding value $\tau$. From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,\Phi_{\sigma})\le c\tau. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $\Phi_{\sigma}$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ \rho(F,\Phi_{\sigma}) =\inf\int\exp(|x-y|/c\tau)\,d\pi(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $\pi$ with marginal distributions $F$ and $\Phi_{\sigma}$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(\tau )$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed.