Cuesta, Juan Antonio; Matrán, Carlos Notes on the Wasserstein metric in Hilbert spaces. (English) Zbl 0688.60011 Ann. Probab. 17, No. 3, 1264-1276 (1989). A necessary condition is established for a pair of random variables which are optimal couplings w.r.t. the \(L^ 2\)-distance. This condition can be motivated by a simple rearrangement argument. It is applied to extend a method of Tanaka for proving the central limit theorem. This method is based on the following observation. If \(X, Y\) are i.i.d. and if \(2^{- 1}(X+Y)\) has the same \(L^ 2\)-distance to the normal distribution (with the same covariance) as \(X\), then \(X\) is normal. Reviewer: L.Rüschendorf Cited in 1 ReviewCited in 25 Documents MSC: 60E05 Probability distributions: general theory 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) Keywords:Wasserstein distance; optimal couplings; central limit theorem PDFBibTeX XMLCite \textit{J. A. Cuesta} and \textit{C. Matrán}, Ann. Probab. 17, No. 3, 1264--1276 (1989; Zbl 0688.60011) Full Text: DOI Euclid