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On the characterization of a class of binary operations on distribution functions. (English) Zbl 0798.60023
The distribution function \(H\) of the sum of two (independent) random variables with distribution functions \(F\) and \(G\), respectively, is the convolution of \(F\) and \(G\). This function \(H\) can be viewed as the result of a binary operation on random variables or as the result of a binary operation on distribution functions. The authors completely characterize certain binary operations on distribution functions; namely, those that are induced pointwise by a function \(\Phi\) and can also be derived from a binary operation \(V\) on random variables. This collection of binary operations is surprisingly small. Indeed in Theorem 1 the authors show that \(\Phi\) must be a quasi copula and \(V\) must be Max, \(\Phi\) must be the dual of a quasi copula and \(V\) must be Min, or \(\Phi\) and \(V\) must both be trivial \((V(x,y)=y\), \(\Phi (a,b) = b\) or \(V(x,y) = x\), \(\Phi (a,b) = a)\). As a result, it follows that certain natural operations or distribution functions such as the mixture of two distributions can not be derived from a binary operation on random variables. Thus binary operations on random variables lead to a corresponding binary operation on distributions, but not conversely.

MSC:
60E99 Distribution theory
60A99 Foundations of probability theory
60E05 Probability distributions: general theory
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[1] Alsina, C.; Schweizer, B., Mixtures are not derivable, Found. phys. lett., 1, 171-174, (1988)
[2] Schweizer, B.; Sklar, A., Operations on distribution functions not derivable from operations on random variables, Studia math., 52, 43-52, (1974) · Zbl 0292.60035
[3] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier New York · Zbl 0546.60010
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