## A remark on associative copulas.(English)Zbl 1010.60014

Let $$I=[0,1]$$. Copulas are distribution functions on $$I^2$$ with uniform marginals. Assume that $$\oplus$$ is a continuous associative operation in $$[0,a]$$, $$a\in [0,\infty ]$$, such that $$t\oplus 0= 0\oplus t=t$$, $$t\oplus a=a\oplus t=a$$ for all $$t\in [0,a]$$. A function $$\psi :[0,a]\to R$$ is called $$\oplus$$-convex if $$\psi (r\oplus t)-\psi (r)\leq \psi (s\oplus t) -\psi (s)$$ for every $$r\leq s$$ and any $$t$$. The main result of the paper can be formulated as follows. Let $$\varphi :I\to [0,a]$$ be a strictly decreasing continuous surjection. Define $$C(x,y) =\varphi ^{-1}(\varphi (x)\oplus \varphi (y))$$. Then $$C$$ is a copula if and only if $$\varphi ^{-1}$$ is $$\oplus$$-convex.

### MSC:

 6e+06 Probability distributions: general theory 6.2e+11 Characterization and structure theory of statistical distributions

### Keywords:

associative copulas; Archimedean copulas
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