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

*As the reader might most likely already finish from the**enthusiastic phrases within the first traces of this assessment, this publication can be**strongly advised to probabilists and statisticians who deal with**distributions with given marginals.***Mededelingen van het Wiskundig Genootschap**

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SCHWEIZER I came across a review of a paper by G. D. Makarov entitled "Estimates for the distribution function of the sum of two random variables with given marginal distributions" [42; MR 83c: 60029]. I looked up the paper and my interest in it increased when I learned that its aim was to answer a question that had been posed by Kolmogorov. But I found Makarov's argument somewhat impenetrable and therefore set out to practice what I have just preached. (z) = Kolmogorov's problem is the following: df(X) = F X and Y with F F such that for all and and df(Y) = G, z in R, infP(X + Y < z) and F(z) = supp(X + Y < z), where the infimum and supremum are taken over all possible joint distribution functions H having margins F and G.

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? The answer is: none. To make this more precise, we need the following: Generally 28 B. 1. f. 's defined on a common probability space, such that df(X) = F, =G df(Y) and df(V(X,Y)) = ~(F,G). 1. Min. Then 'T Let T be any (left-continuous) t-norm other than is not derivable from any function on random variables.

F) If f and g are strictly monotone respectively. f. of X and relation coefficient r. function of Irl. Y) l£1 2 . v. f. 's H sequence n {H } Ran Y. is bivariate normal. Y) is a strictly increasing n . 2 •... , and are pairs of continuous respectively. Y). If and A Y then a. s. Y). Y) n n = a(x. Y) • It follows from property (F) that a is a measure of monotone de- pendence. • a rank statistic. The properties (A) - (H) differ from Renyi's original conditions. Renyi did not require the continuity condition (H).