Characterization of idempotent 2-copulas


A 2-copula A induces a transition probability function p_A via

where S\in \cal B, \cal B denoting the Lebesgue measurable subsets of[0,1].  We say that a set S is invariant under A if p_A(x,S)=\chi _S(x)for almost all x\in [0,1], \chi _S being the characteristic function ofS.  The sets S invariant under A form a sub-\sigma-algebra of theLebesgue measurable sets, which we denote {\cal B}_A. A set S\in {\cal B}_Ais called an atom if it has positive measure and if for any S'\in {\cal B}_A,\lambda (S'\cap S) is either \lambda (S) or 0.
A 2-copula F is idempotent if F*F=F. Here * denotes the product defined in [1]. Idempotent 2-copulas are classified and characterized asfollows:
(i) An idempotent F is said to be nonatomic if {\cal B}_F contains noatoms.  If F is a nonatomic idempotent, then it is the product of a leftinvertible copula and its transpose.  That is, there exists a copula B such that
where M(x,y)=\min(x,y).
(ii) An idempotent F is said to be totally atomic if there exist essentiallydisjoint atoms S_n\in {\cal B}_F with
If F is a totally atomic idempotent, then it is conjugate to an ordinal sumof copies of the product copula.  That is, there exists a copula C satisfyingC*C^T=C^T*C=M and a partition \cal P of [0,1] such that
\begin{equation}F=C*(\oplus _{\cal P}F_k)*C^T \end{equation} where eachcomponent F_k in the ordinal sum is the product copula P.
(iii) An idempotent F is said to be atomic (but not totally atomic) if {\cal  B}_F contains atoms but the sum of the measures of a maximal collection ofessentially disjoint atoms is strictly less than 1.  In this mixed case, thereexists a copula C invertible with respect to M and a partition \cal P of[0,1] for which (1) holds, with F_1 being a nonatomic idempotent copula andwith F_k=P for k>1.
Some of the immediate consequences of this characterization are discussed.

DOI Code: 10.1285/i15900932v30n1p147

copula; idempotent; star product

Classification: 60G07; 60J05; 60J25

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