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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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##### References:
 [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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