Parametrizing Doubly Stochastic Measures

Doubly stochastic measures can be identi ed with the trace of a pair of (Lebesgue) measure preserving maps of the unit interval to itself. It has been of traditional interest in probability theory to produce a random vector (or metric space element), which has a given distribution and is de ned on a standard space, such as [0; 1] endowed with Lebesgue measure. In a classic work, L evy [4, section 23] used an approach based on conditioning. For the purpose of the Skorokhod representation, Billingsley [1, Theorem 3.2] considered the case of random elements of a general metric space. Whitt [11, Lemma 2.7] considered general measures on R n and employed a Borel isomorphism to treat questions of extremal correlation and minimal variance. R uschendorf [8] used a similar approach to consider a general class of optimization problems. In this note, we re-visit the question of a representing random element in the special case of a doubly stochastic measure on the unit square. First we show the existence of a random element (using essentially Whitt's approach) with a re nement to a canonical representation. Then we turn to criteria for extremality of a doubly stochastic measure. Our aim is to provide the reader who is interested in extremality with di erent-looking settings. This paper was invited for presentation at the AMS-IMS-SIAM Joint Summer Research Conference on Distributions with Fixed Marginals, Doubly-Stochastic Measures, Supported in part by ONR Grant N00014-90-J-1641 and NSF Grant DMS-9002665. Permanent address: Department of Statistics, Box U-120, University of Connecticut, Storrs, CT 06269-3120. and Markov Operators, July 31-August 6, 1993. At the Conference, the author learned of A Representation for Doubly Stochastic Measures by W.F. Darsow and E.T. Olsen. Subsequently, a revision was provided to the author. Interested readers are directed to this work for variant arguments and related topics, including copulas. 1 Doubly Stochastic Measures via Pairs of Measure-