The Annals of Mathematical Statistics

The Existence of Probability Measures with Given Marginals

V. Strassen

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Abstract

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.

Article information

Source
Ann. Math. Statist. Volume 36, Number 2 (1965), 423-439.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177700153

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aoms/1177700153

Mathematical Reviews number (MathSciNet)
MR177430

Zentralblatt MATH identifier
0135.18701

Citation

Strassen, V. The Existence of Probability Measures with Given Marginals. The Annals of Mathematical Statistics 36 (1965), no. 2, 423--439. doi:10.1214/aoms/1177700153. http://projecteuclid.org/euclid.aoms/1177700153.


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