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}\|\leτ\}=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 $τ$. From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,Φ_σ)\le cτ. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $Φ_σ$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ ρ(F,Φ_σ) =\inf\int\exp(|x-y|/cτ)\,dπ(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $π$ with marginal distributions $F$ and $Φ_σ$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(τ)$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed.