Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances
For two Borel probability measures \(\mu\) and \(\gamma\) on \(\mathbb{R}\) with finite absolute moments of order \(p\geq1\), the power transport distance of order \(p\) is defined by \[ W_p(\mu,\gamma)=\inf_\pi\left(\int_{-\infty}^\infty\int_{-\infty}^\infty|x-y|^p\,d\pi(x,y)\right)^{1/p}\,, \] where the infimum is taken over all probability measures \(\pi\) on \(\mathbb{R}^2\) with marginal distributions \(\mu\) and \(\gamma\). The main results of the present paper are bounds on the distance \(W_p(\mu_n,\gamma)\), where \(\mu_n\) is the distribution of a sum of appropriately normalized independent random variables \(X_1+\cdots+X_n\), and \(\gamma\) is the standard normal law. In particular, the author proves that for any \(p\geq1\) we have \[ W_p(\mu_n,\gamma)\leq c_pL_{p+2}^{1/p}\,, \] where \(L_{p+2}=\sum_{k=1}^n\mathbb{E}|X_k|^{p+2}\) is the Lyapunov coefficient, and \(c_p\) is a constant depending on \(p\). This extends results of \textit{E. Rio} [Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 3, 802--817 (2009; Zbl 1175.60020)], who considered the case \(1\leq p\leq2\). The author also establishes a refinement of his bounds which holds in the case of identically distributed summands \(X_1,\ldots,X_n\) satisfying Cramér's condition. The proofs employ Edgeworth expansions and a certain signed measure on the real line which acts as a `correction' to the normal law.
scientific article; zbMATH DE number 6837238
scientific article