Cuesta-Albertos, J. A.; Matrán-Bea, C.; Tuero-Diaz, A. On lower bounds for the \(L^ 2\)-Wasserstein metric in a Hilbert space. (English) Zbl 0870.60005 J. Theor. Probab. 9, No. 2, 263-283 (1996). Summary: We provide two families of lower bounds for the \(L^2\)-Wasserstein metric in separable Hilbert spaces which depend on the basis chosen for the space. Then we focus on one of these families and we provide a necessary and sufficient condition for the supremum in it to be attained. In the finite-dimensional case, we identify the basis which provides the most accurate lower bound in the family. Cited in 1 ReviewCited in 15 Documents MSC: 60B05 Probability measures on topological spaces Keywords:Wasserstein metric; lower bound; Gaussian distributions; Hilbert spaces PDFBibTeX XMLCite \textit{J. A. Cuesta-Albertos} et al., J. Theor. Probab. 9, No. 2, 263--283 (1996; Zbl 0870.60005) Full Text: DOI References: [1] Bickel, P. J., and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap.Ann. Statist. 9, 1196–1217. · Zbl 0472.62054 [2] Conway, J. B. (1985).A Course in Functional Analysis, Springer, New York. · Zbl 0558.46001 [3] Cuesta-Albertos, J. A., Dominguez-Menchero, J. S., and Matran-Bea, C. (1991). Some stochastics on monotone functions.J. Comp. Applied Math. 55, 165–182. · Zbl 0827.62034 [4] Cuesta-Albertos, J. A., and Matran-Bea, C. (1991). Notes on the Wasserstein metric in Hilbert spaces.Ann. Prob. 17, 1264–1276. · Zbl 0688.60011 [5] Cuesta-Albertos, J. A., C. Matran-Bea, L., and Tuero-Diaz, A. (1993). Optimal maps for theL 2-Wasserstein distance. Preprint. [6] Cuesta-Albertos, J. A., Rüschendorf, L., and Tuero-Diaz, A. (1993). Optimal coupling of multivariate distributions and stochastic processes.J. Multivariate Anal. 46, 355–361. · Zbl 0788.60025 [7] Gelbrich, M. (1990). On a formula for theL 2-Wasserstein metric between measures on Euclidean and Hilbert Spaces.Math. Nachr. 147, 185–203. · Zbl 0711.60003 [8] Knott, M., and Smith, C. S. (1984). On the optimal mapping of distributions.J. Optim. Theory Appl. 43, 39–49. · Zbl 0519.60010 [9] Laha, R. G., and Rohatgi, V. K. (1979).Probability Theory, J. Wiley and Sons, Chichester. [10] Olkin, I., and Pukelsheim, F. (1982). The distance between two random vectors with given dispersion matrices.Linear Algebra Appl. 48, 257–263. · Zbl 0527.60015 [11] Rüschendof, L., and Rachev, S. T. (1990). A characterization of random variables with minimumL 2-distance.J. Multivariate Anal. 32, 48–54. · Zbl 0688.62034 [12] Rüschendorf, L. (1991). Fréchet bounds and their applications. In Dall’Aglio, G., Kotz, S., and Salinetti (eds.),Advances in probability distributions with given marginals. pp. 151–187. [13] Schweizer, B. (1991). Thirty years of copulas. In Dall’Aglio, G., Kotz, S., and Salinetti, G. (eds.),Advances in Probability Distributions with Given Marginals. Pp. 13–50. · Zbl 0727.60001 [14] Smith, C., and Knott, M. (1990). A note on the bound for theL 2 Wasserstein Metric. (Unpublished paper). [15] Stout, W. F. (1974).Almost Sure Converge, Academic Press, New York. · Zbl 0321.60022 [16] Vakhania, N. N., Tarieladze, V. I., and Chobanyan, S. A. (1987).Probability Distributions on Banach Spaces, Reidel, Dordrecht. · Zbl 0698.60003 [17] Whitt, W. (1976). Bivariate distributions with given marginals.Ann. Statist. 4, 1280–1289. · Zbl 0367.62022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.