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April, 1965 The Existence of Probability Measures with Given Marginals
V. Strassen
Ann. Math. Statist. 36(2): 423-439 (April, 1965). DOI: 10.1214/aoms/1177700153


First an integral representation of a continuous linear functional dominated by a support function in integral form is given (Theorem 1). From this the theorem of Blackwell-Stein-Sherman-Cartier [2], [20], [4], is deduced as well as a result on capacities alternating of order 2 in the sense of Choquet [5], which includes Satz 4.3 of [23] and a result of Kellerer [10], [12], under somewhat stronger assumptions. Next (Theorem 7), the existence of probability distributions with given marginals is studied under typically weaker assumptions, than those which are required by the use of Theorem 1. As applications we derive necessary and sufficient conditions for a sequence of probability measures to be the sequence of distributions of a martingale (Theorem 8), an upper semi-martingale (Theorem 9) or of partial sums of independent random variables (Theorem 10). Moreover an alternative definition of Levy-Prokhorov's distance between probability measures in a complete separable metric space is obtained (corollary of Theorem 11). Section 6 can be read independently of the former sections.


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V. Strassen. "The Existence of Probability Measures with Given Marginals." Ann. Math. Statist. 36 (2) 423 - 439, April, 1965.


Published: April, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0135.18701
MathSciNet: MR177430
Digital Object Identifier: 10.1214/aoms/1177700153

Rights: Copyright © 1965 Institute of Mathematical Statistics

Vol.36 • No. 2 • April, 1965
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