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Lipschitz continuity of copulas w.r.t. \(L_p\)-norms. (English) Zbl 1189.26015
Binary aggregation functions, i.e. increasing functions \(A : [0, 1]^2 \rightarrow [0, 1]\), satisfying the conditions \(A(0,0)=0\), \(A(1,1)=1\), are of interest in the paper. The smallest and greatest binary aggregation functions, which are \(1-p\)-Lipschitz, are Yager’s triangular norm
\[ T_p^Y(x,y)= \begin{cases} \max\{0,1-((1-x)^p+ (1-y)^p)^{1/p}\}, &\text{if }p<\infty,\\ \min \{x, y\}, &\text{if }p=\infty, \end{cases} \] and triangular conorm
\[ S_p^Y(x,y)= \begin{cases} \min\{1, (x^p+y^p)^{1/p}\}, &\text{if }p<\infty,\\ \max\{x,y\}, &\text{if }p= \infty, \end{cases} \] respectively, as it was shown earlier.
Special class of aggregation functions form so called copulas, i.e., those functions \(C:[0,1]^2\to[0,1]\) which have the neutral element 1 and are 2-increasing, i.e., for the \(C\)-volume \(V_C([x,x']\times [y,y']\) of the rectangle \([x,x']\times [y,y']\) the following inequality holds:
\[ V_C([x,x']\times [y,y']= C(x',y')+ C(x,y)- C(x,y')- C(x',y)\geq 0, \] for any \(0\leq x\leq x'\) and \(0\leq y\leq y'\).
It is shown that several well-known construction methods yield an \(1-p\)-Lipschitz copula when departing from copulas of the same type. The main part of the paper is devoted to the study of properties of important classes of copulas, namely, Archimedean copulas, extreme value copulas and Archimax copulas.

26B05 Continuity and differentiation questions
26B15 Integration of real functions of several variables: length, area, volume
Full Text: DOI
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