# Advances in Probability Distributions with Given Marginals: by G. Dall’aglio (auth.), G. Dall’Aglio, S. Kotz, G. Salinetti

By G. Dall’aglio (auth.), G. Dall’Aglio, S. Kotz, G. Salinetti (eds.)

As the reader could most likely already finish from theenthusiastic phrases within the first traces of this evaluation, this publication can bestrongly steered to probabilists and statisticians who deal withdistributions with given marginals.
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5. DERIVABILITY AND BOUNDS We conclude the first part of this paper with a discussion of several additional results that were obtained in the 1970's in the course of our work on probabilistic metric spaces. f. f. 's, Y such that (or, as we shall henceforth often write, df(X) df(X + Y) and = F* G. Y such that df(X) df(L(X,Y)) = F, = aC,L. v. 's is the copula of Since the binary operations TT X and X and Y, playa promi- nent role in the theory of probabilistic metric spaces, and since TMin = a Min , it is natural to ask: What binary operations on random variables correspond to these operations?

It Hoeffding's collected papers THIRTY YEARS OF COPULAS 35 Since 1959, copulas have been rediscovered by several authors. first to do so were Kimeldorf and Sampson. 9) to define two-dimensional copulas. They called them uniform representations; and in [35] and several subsequent papers [36, 37] they developed many of their basic properties and used them as a tool to define and study various dependence notions. Further de- tails and additional references may be found in Sampson's contribution to this volume.

For example, if for any a > 0, we let L be defined by L (u,v) a a a (u + va)l/a, then the operations are the a-convolutions C,La which have been extensively studied by K. Urbanik [73]. 13) TT ,L (F ,G)(x) ~+ belonging to yield a two-parameter family of defined on TT,L When restricted to B of L V via sUPL(u,v) = xT(F(u) ,G(v)). n V, this leads to a larger class of triangle functions (see Chapter 7 of [62] for details). J. Frank undertook a detailed study of the operations 0c [15]. From the point of view of probabilistic metric spaces, where we are interested in obtaining a supply of triangle functions, it is natural to ask: When is associative?

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