Notes on the Wasserstein metric in Hilbert spaces (Q1263861)

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.