A 2-copula
induces a transition probability function
via
![](http://212.189.136.205/plugins/generic/latexRender/cache/4141133971826b5656e566423ab4576f.png)
where
![S\in \cal B](http://212.189.136.205/plugins/generic/latexRender/cache/0facb01c50351de791f37da9b04f0cd1.png)
,
![\cal B](http://212.189.136.205/plugins/generic/latexRender/cache/06201a12c93c4f2d041f50136750c5cf.png)
denoting the Lebesgue measurable subsets of
![[0,1]](http://212.189.136.205/plugins/generic/latexRender/cache/ccfcd347d0bf65dc77afe01a3306a96b.png)
. We say that a set
![S](http://212.189.136.205/plugins/generic/latexRender/cache/5dbc98dcc983a70728bd082d1a47546e.png)
is invariant under
![A](http://212.189.136.205/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
if
![p_A(x,S)=\chi _S(x)](http://212.189.136.205/plugins/generic/latexRender/cache/c07c70b33aade7f609bdd7323be856b2.png)
for almost all
![x\in [0,1]](http://212.189.136.205/plugins/generic/latexRender/cache/c628ba2b1047de93f66cb815d986e107.png)
,
![\chi _S](http://212.189.136.205/plugins/generic/latexRender/cache/3973dbce0fe19f7f96c978624ee625f0.png)
being the characteristic function of
![S](http://212.189.136.205/plugins/generic/latexRender/cache/5dbc98dcc983a70728bd082d1a47546e.png)
. The sets
![S](http://212.189.136.205/plugins/generic/latexRender/cache/5dbc98dcc983a70728bd082d1a47546e.png)
invariant under
![A](http://212.189.136.205/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
form a sub-
![\sigma](http://212.189.136.205/plugins/generic/latexRender/cache/a2ab7d71a0f07f388ff823293c147d21.png)
-algebra of theLebesgue measurable sets, which we denote
![{\cal B}_A](http://212.189.136.205/plugins/generic/latexRender/cache/39abb13c238d0973e5c8ee5111675eb0.png)
. A set
![S\in {\cal B}_A](http://212.189.136.205/plugins/generic/latexRender/cache/2130a1d693a8b68a7bf87275a40f519f.png)
is called an atom if it has positive measure and if for any
![S'\in {\cal B}_A](http://212.189.136.205/plugins/generic/latexRender/cache/2e7ad2c8be10aaed5d8d19da9b9f537d.png)
,
![\lambda (S'\cap S)](http://212.189.136.205/plugins/generic/latexRender/cache/2218cb527efdcefdd9f848d6bac3b93a.png)
is either
![\lambda (S)](http://212.189.136.205/plugins/generic/latexRender/cache/1afe6f8ef0883636f4d1be3d7cbf9f61.png)
or 0.
A 2-copula
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
is idempotent if
![F*F=F](http://212.189.136.205/plugins/generic/latexRender/cache/08626eb37ca8c2036f36db573bf460d7.png)
. Here
![*](http://212.189.136.205/plugins/generic/latexRender/cache/3389dae361af79b04c9c8e7057f60cc6.png)
denotes the product defined in [1]. Idempotent 2-copulas are classified and characterized asfollows:
(i) An idempotent
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
is said to be nonatomic if
![{\cal B}_F](http://212.189.136.205/plugins/generic/latexRender/cache/9ab77610a6b9f323acd5e8aba68a7555.png)
contains noatoms. If
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
is a nonatomic idempotent, then it is the product of a leftinvertible copula and its transpose. That is, there exists a copula
![B](http://212.189.136.205/plugins/generic/latexRender/cache/9d5ed678fe57bcca610140957afab571.png)
such that
![](http://212.189.136.205/plugins/generic/latexRender/cache/bb560edd8fa2685cc468e2e9cc835204.png)
![](http://212.189.136.205/plugins/generic/latexRender/cache/916f3d2a48a3b3ed23bc9f7a1cacaf77.png)
where
![M(x,y)=\min(x,y).](http://212.189.136.205/plugins/generic/latexRender/cache/8d5ef11623103704b2e878af05ecf619.png)
(ii) An idempotent
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
is said to be totally atomic if there exist essentiallydisjoint atoms
![S_n\in {\cal B}_F](http://212.189.136.205/plugins/generic/latexRender/cache/ae301c70662c36c6e11a261f763026d4.png)
with
![](http://212.189.136.205/plugins/generic/latexRender/cache/ff49373d6f99f19a5a1f26bd1ba3faf3.png)
If
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
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](http://212.189.136.205/plugins/generic/latexRender/cache/0d61f8370cad1d412f80b84d143e1257.png)
satisfying
![C*C^T=C^T*C=M](http://212.189.136.205/plugins/generic/latexRender/cache/72e050c2710284684451de727d46b578.png)
and a partition
![\cal P](http://212.189.136.205/plugins/generic/latexRender/cache/f2790946d2c2f91343e49fd8ca3c9e94.png)
of
![[0,1]](http://212.189.136.205/plugins/generic/latexRender/cache/ccfcd347d0bf65dc77afe01a3306a96b.png)
such that
\begin{equation}F=C*(\oplus _{\cal P}F_k)*C^T \end{equation} where eachcomponent
![F_k](http://212.189.136.205/plugins/generic/latexRender/cache/9353d3f34be4a9b672be4303774ad527.png)
in the ordinal sum is the product copula
![P](http://212.189.136.205/plugins/generic/latexRender/cache/44c29edb103a2872f519ad0c9a0fdaaa.png)
.
(iii) An idempotent
![F](http://212.189.136.205/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
is said to be atomic (but not totally atomic) if
![{\cal B}_F](http://212.189.136.205/plugins/generic/latexRender/cache/d464953fd0e90f52db208050b5d8e31c.png)
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](http://212.189.136.205/plugins/generic/latexRender/cache/0d61f8370cad1d412f80b84d143e1257.png)
invertible with respect to
![M](http://212.189.136.205/plugins/generic/latexRender/cache/69691c7bdcc3ce6d5d8a1361f22d04ac.png)
and a partition
![\cal P](http://212.189.136.205/plugins/generic/latexRender/cache/f2790946d2c2f91343e49fd8ca3c9e94.png)
of
![[0,1]](http://212.189.136.205/plugins/generic/latexRender/cache/ccfcd347d0bf65dc77afe01a3306a96b.png)
for which (1) holds, with
![F_1](http://212.189.136.205/plugins/generic/latexRender/cache/bc6b0efd3bed4dfabe15757cf4089d87.png)
being a nonatomic idempotent copula andwith
![F_k=P](http://212.189.136.205/plugins/generic/latexRender/cache/d4aa0e14409c830f4a1c570e5128fb33.png)
for
![k>1](http://212.189.136.205/plugins/generic/latexRender/cache/d37c0d918e9117937016dc3dd13c2faa.png)
.
Some of the immediate consequences of this characterization are discussed.