Characterization of idempotent 2-copulas

Abstract

A 2-copula $A$ induces a transition probability function $p_A$ via$$ p_A(x,S)=\frac {d }{d x}\int _S \frac {\partial }{\partial t}A(x,t)\,dt.$$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 of$S$.  The sets $S$ invariant under $A$ form a sub-$\sigma$-algebra of theLebesgue measurable sets, which we denote ${\cal B}_A$. A set $S\in {\cal B}_A$is 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 $$B*B^T=F, \qquad \text{\rm and}\quad $$ $$B^T*B=M,$$ 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$$\sum _n\lambda (S_n)=1. $$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 $C$ satisfying$C*C^T=C^T*C=M$ 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.