Summary
By an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases. For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’. We give some applications to the problem of approximation of stochastic processes by simpler ones, as e.g. weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximation.
Article PDF
Similar content being viewed by others
References
Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7, 29–54 (1979)
Cambanis, S., Simons, G., Stout, W.: Inequalities for Ek(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitstheor. verw. Geb. 36, 285–294 (1976)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. New York-Heidelberg-New York: Springer 1977
Dall'Aglio, G.: Sugli estremi di momenti delle funzioni di ripartizione doppia. Annali Scuola Normale Superiore di Pisa, Vol. 10, 35–74 (1956)
Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15, 458–486 (1970)
Dowson, D.C., Landau, B.V.: The Fréchet distance between multivariate normal distributions. J. Multivariate Anal. 12, 450–455 (1982)
Eberlein, E.: Strong approximation of very weak Bernoulli processes. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 62, 17–37 (1983)
Eberlein, E.: An invariance principle for lattices of dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 50, 119–133 (1979)
Gordin, M.I.: The central limit theorem for stationary processes. Sov. Math. Dokl. 10, 1174–1176 (1969)
Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of Random Variables. Groningen: Wolks Noordhoff 1971
Kantorovic, L., Rubinstein, G.: On a space of completely additive functions. Vestn. Leningr. Univ. Math. 13, 7, 52–59 (1958)
Major, P.: On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8, 487–517 (1978)
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. New York: Academic Press 1979
Olkin, L, Pukelsheim, F.: The distance between two random vectors with given dispersion matrizes. Linear Algebra Appl. 43, 257–263 (1982)
Philipp, W., Stout, W.: Almost sure invariance principles for sums of weakly dependent random variables. Am. Math. Soc., Mémoir 161.
Rachev, S.T.: Minimal metrices in the random variable space. Publ. Inst. Stat. Univ. Paris 28, 1–26 (1982)
Rüschendorf, L.: Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–62 (1983)
Rüschendorf, L.: Robust tests for independence. Preprint 1981
Schwarz, G.: Finitely determined processes — an indiscrete approach. J. Math. Anal. Appl. 76, 146–158 (1980)
Strittmatter, W.: Metriken für stochastische Prozesse und schwache Abhängigkeit. Diplomarbeit, Freiburg 1982
Statulevicius, V.A.: Limit theorems for sums of random variables related to a markov chain II. Litov. Mat. Sb. 9, 635–672 (1969)
Vallander, S.S.: Calculation of the Wasserstein distance between distributions on the line. Theory Probab. Appl. 18, 784–786 (1973)
Volkonskii, V.A., Rozanov, J.A.: Some limit theorems for random functions II. Theory Probab. Appl. 6, 186–198 (1961)
Zolotarev, V.M.: Probability metrics. Theory Probab. Appl. 28, 278–302 (1983)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rüschendorf, L. The Wasserstein distance and approximation theorems. Z. Wahrscheinlichkeitstheorie verw Gebiete 70, 117–129 (1985). https://doi.org/10.1007/BF00532240
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00532240