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BY 4.0 license Open Access Published by De Gruyter Open Access November 14, 2023

Characterization of pre-idempotent Copulas

  • Wongtawan Chamnan and Songkiat Sumetkijakan EMAIL logo
From the journal Dependence Modeling

Abstract

Copulas C for which ( C t C ) 2 = C t C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of . Identifying copulas with Markov operators on L 2 , the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

1 Introduction

As is well-known, dependence between two continuous random variables can be captured by their copula. Stochastic independence is modeled by the product copula Π . Monotonic dependence is modeled by the Fréchet-Hoeffding bounds M C e , e (comonotonic) or W C e , 1 e (countermonotonic) and a general complete dependence copula can be represented by a measure-preserving transformation (MPT) f on [ 0 , 1 ] and written as C e , f or C f , e , where e denotes the identity function. To cite an instance, C e , f is the copula of [ 0 , 1 ] -uniformly distributed random variables U and V for which V = f ( U ) almost surely, and the mass of its associated measure is concentrated on the graph of f [6,12]. See equation (2.3) for the definition of C f , g . Even though every copula is of the form C f , g = C f , e C e , g , the Markov product of C f , e and C e , g and relations between the copula and the MPTs f and g are far from simple [5,13,23]. Swapping places of the factors defines a factorizable copula C e , f C g , e , whose associated measure has its mass concentrated on { ( x , y ) : f ( x ) = g ( y ) } [16,19]. This implies, in particular, that factorizable copulas are singular, yet at the same time gives nice, vivid, and potentially useful interpretations of factorizable copulas. In fact, C e , f C g , e is the copula of some implicitly dependent U and V for which f ( U ) = g ( V ) almost surely. See the studies by Panyasakulwong et al. [14,15] for novel investigations on such implicit dependence copulas in which it was shown that the converse is not always true. While complete dependence copulas are at least one-way deterministic, factorizable copulas can be much less deterministic, especially when f and g are highly noninjective. Surprisingly, complete dependence copulas are quite ubiquitous in the theory of copulas [22]. Being more stochastic, factorizable copulas could conceivably play a complementing role to complete dependence copulas. Thus, a systematic study of factorizable copulas must be commenced.

In 2010, Darsow and Olsen [4] provided a characterization of idempotent copulas by splitting them into cases according to the atomicity types. For instance, nonatomic idempotent copulas are precisely the symmetric factorizable copulas C e , f C f , e , and every totally atomic idempotent copula is conjugate to an ordinal sum of a countable copies of Π . Even though both M and Π are idempotent, all other complete dependence copulas are not. Moreover, idempotent copulas must be symmetric [4,21] and hence are too limited to model general dependence. However, a novel partial ordering on the idempotent copulas was shown [4] to be equivalent to the inclusion on the classes of their invariant sets. Through some examples, this ordering is related to degrees of dependence. Our goal is to construct and study a class of copulas that accommodates sufficiently many dependence structures, or rather dependence scaffolding in the sense that the class contains all factorizable copulas while still possesses adequate properties to gauge dependence levels. Clearly, for every factorizable copula C = C e , f C g , e , the transpose-product C t C is an idempotent, and so is C C t . Hence, we shall investigate the class of pre-idempotent copulas defined as copulas C for which C t C is idempotent. The class contains not only idempotent copulas and factorizable copulas but also conjugates of ordinal sums of factorizable copulas and copies of Π . Here, C is a conjugate of D if C = C r , e D C e , p for some invertible MPTs r and p . We obtain a characterization of pre-idempotent copulas using tools from the precedent characterization of idempotent copulas in the study by Darsow and Olsen [4] and the fact that the set of pre-idempotent copulas and the set of partially isometric Markov operators are isomorphic. In addition, the atomicity of the associated σ -algebras of copulas defined in the study by Sumetkijakan [19] plays the main role in splitting our consideration into cases. A copula C is in if and only if C is conjugate to an ordinal sum D P D k of a factorizable copula and/or copies of Π as follows:

  • if C is nonatomic, then P = { [ 0 , 1 ] } and D 1 is factorizable, which implies that C is factorizable;

  • if C is totally atomic, then D k = Π for all k ; and

  • if C is atomic but not totally atomic, then D 1 is factorizable and D k = Π for all k 2 .

The above characterization and its proof are in Section 3. In the last section, we consider the set of copulas under the strong topology induced by the modified Sobolev norm defined in equation (2.4). It is shown that is closed, and the set of factorizable copulas is dense in .

2 Background knowledge and tools

Throughout the manuscript, let B B ( I ) denote the Borel σ -algebra on I [ 0 , 1 ] , and λ the Lebesgue measure on I . The class of essentially bounded Borel measurable functions on I is denoted by L . For 1 p < , L p denotes the class of p -integrable Borel measurable functions on I , each of which is a Banach space under the usual L p -norm p . Clearly, L L p L 1 . The indicator function of a Borel set A I is denoted by 1 A . The identity function on I is denoted by e .

Let p [ 1 , ) and q ( 1 , ] be its conjugate exponent, i.e., 1 p + 1 q = 1 . For a bounded linear operator S on L p , its adjoint is the unique bounded linear operator S * on L q that satisfies I S ψ ξ d λ = I ψ S * ξ d λ for all ψ L p and ξ L q . Recall that S * = S . The case p = 2 is special in that S * and S are both linear operators on the same space L 2 .

2.1 Markov operators

A linear operator S : L p L p is called a Markov operator on L p if

  1. S 1 I = 1 I ;

  2. I S ψ d λ = I ψ d λ for ψ L p ; and

  3. S ψ 0 whenever ψ 0 .

The class of all such operators is denoted by p . By the linearity and positivity of S p , S ψ S ψ for all ψ L p . For S 1 , it holds that for ψ L p and ξ L ,

(2.1) S ψ 1 = I S ψ d λ I S ψ d λ = I ψ d λ = ψ 1

and

(2.2) S ξ S ξ S ξ = ξ S 1 I = ξ 1 I = ξ .

Thus, S is bounded as an operator on L 1 and as an operator on L with norm 1. For 1 < p < , by Riesz-Thorin interpolation theorem [7, Theorem 8.23], S ψ p ψ p for all ψ L p ; hence, every S 1 restricts to a Markov operator on L p . Since a Markov operator on L extends uniquely to a Markov operator on L 1 , the adjoint of S 1 can and usually will be regarded as the adjoint Markov operator S * 1 (see [7, Theorem 13.2]). Moreover, for S , T 1 , ( S * ) * = S , T S 1 and ( T S ) * = S * T * . For 1 p < , the strong operator topology (SOT) on p is the coarsest topology such that, for each fixed ψ L p , the evaluation map p L p , S S ψ is continuous. The weak operator topology (WOT) on p is the coarsest topology such that, for each fixed ψ L p and ξ L q , the evaluation map p R , S I S ψ ξ d λ is continuous.

The following proposition quoted from [7, Proposition 13.6] says that 1 and p are homeomorphic. Its proof is a good exercise in functional analysis.

Proposition 2.1

For 1 p , the restriction mapping Φ p : 1 p defined by Φ p ( S ) S L p is a bijection. If p < and q is its conjugate exponent, then Φ q ( S * ) = Φ p ( S ) * for S 1 and the mapping Φ p is a homeomorphism for the weak as well as the SOTs.

For a linear operator T on L 2 , recall that T is called an isometry if T φ 2 = φ 2 for all φ L 2 , and it is called a partial isometry if it acts isometrically on the orthogonal complement of its kernel, i.e., T φ 2 = φ 2 for all φ ker ( T ) . A bounded linear operator P on L 2 is called a projection if it is self-adjoint and P 2 = P . Let us quote a characterization of partial isometries from the study by Fernández-Polo and Peralta [8] and Garcia et al. [9], which is an important tool in our investigation of pre-idempotent copulas.

Proposition 2.2

For a bounded linear operator T on L 2 , the following are equivalent:

  1. T is a partial isometry.

  2. T * is a partial isometry.

  3. T = T T * T .

  4. T * = T * T T * .

  5. T * T is a projection.

  6. T T * is a projection.

2.2 Copulas

A function C : I 2 I is said to be a copula if, for u , v , u , v I ,

  1. C ( u , 0 ) = C ( 0 , v ) = 0 ;

  2. C ( u , 1 ) = u and C ( 1 , v ) = v ; and

  3. C ( u , v ) C ( u , v ) C ( u , v ) + C ( u , v ) 0 whenever u u and v v .

The class of copulas is denoted by C . Every C C induces a unique doubly stochastic measure μ C via μ C ( ( a , b ] × ( c , d ] ) C ( b , d ) C ( b , c ) C ( a , d ) + C ( a , c ) and the support of C is defined as the support of the induced measure μ C , i.e., supp ( C ) I 2 \ { O I 2 : O is open and μ C ( O ) = 0 } . Denote C t ( u , v ) C ( v , u ) for u , v I ; and a copula C is said to be symmetric if C t = C . The most well-known copulas are the product copula Π ( u , v ) u v and the Fréchet-Hoeffding upper and lower bounds M ( u , v ) min { u , v } and W ( u , v ) max { 0 , u + v 1 } , respectively. The Markov product, also called the *-product, is a binary operation on C , which is defined as follows:

( A B ) ( u , v ) 0 1 2 A ( u , t ) 1 B ( t , v ) d t ,

under which the class C is a monoid having M as the identity and Π as the zero. A copula C is said to be left (respectively, right) invertible if there is a copula D for which D C = M (respectively, C D = M ); and D , which must be C t if exists, is called a left (respectively, right) inverse of C . C is said to be invertible if it is both left and right invertible. Note that ( A B ) t = B t A t .

Let be the set of all measure-preserving functions on ( I , B , λ ) . A function f is said to possess an essential inverse g if g f = e = f g a.e. We denote by inv the set of measure-preserving functions that possess essential inverses. Define the copula C f , g induced by f and g in by [6]

(2.3) C f , g ( u , v ) λ ( f 1 ( [ 0 , u ] ) g 1 ( [ 0 , v ] ) ) for u , v I .

The left (respectively, right) invertible copulas are exactly the copulas C e , f (respectively, C f , e ) for f . Note that C f , e C e , f = M and that the invertible copulas are exactly the copulas C e , h , where h inv .

From Olsen et al. [13], there is a one-to-one correspondence between C and 1 via the isomorphism C T C defined by ( T C ψ ) ( x ) d d x 0 1 2 C ( x , t ) ψ ( t ) d t for x I . In particular, T C D = T C T D for C , D C . Also, the adjoint of T C is T C * = T C t and hence T C * * = T C . For brevity, we denote T f , g T C f , g . From the study by Darsow and Olsen [4], for every f and ϕ L 1 , T e , f ( ϕ ) = ϕ f and T f , e ( ϕ f ) = ϕ . If T * 1 is the adjoint of T 1 , then, by Proposition 2.1, ( T L 2 ) * = ( Φ 2 ( T ) ) * = Φ 2 ( T * ) = T * L 2 , and so we usually write the corresponding L 2 -Markov operators of C and its adjoint as T C and T C * , respectively.

In 1992, Darsow et al. [2] showed that, with respect to the uniform distance d , the class C is closed and compact and the class of invertible copulas is dense in C . Let { A n } and { B n } be sequences in C converging to copulas A and B , respectively. It holds that A n B A B and B A n B A , but it is not always true that A n B n A B .

Let S denote a modified Sobolev norm on span ( C ) defined by (see [3, the norm 1 , 2 ] and [18])

(2.4) C S 2 0 1 0 1 1 C ( u , v ) 2 + 2 C ( u , v ) 2 d u d v .

Let d 2 be the metric on C induced by the modified Sobolev norm, i.e., d 2 ( A , B ) A B S . It follows from symmetry that A n d 2 A if and only if A n t d 2 A t . It was shown in the study by Darsow and Olsen [3] that the Markov product is jointly continuous with respect to d 2 and that the metric space ( C , d 2 ) is complete. The metric D 2 introduced in the study by Trutschnig [20] is defined by D 2 2 ( A , B ) 0 1 0 1 1 ( A B ) ( u , v ) 2 d u d v . It can be regarded as an asymmetric version of d 2 via the relation d 2 2 ( A , B ) = D 2 2 ( A , B ) + D 2 2 ( A t , B t ) . The topology induced by d 2 coincides [20] with the topology on C induced by the -convergence studied in the works of Mikusiński and Taylor [10,11]. As such, the corresponding topology on 1 shall be denoted by O . Let us denote, respectively, by O w and O s the WOT and the SOT on 1 . Using the one-to-one correspondence between C and 1 , the topology O w on 1 corresponds to the topology on C induced by the metric d (see [13]), while it was shown in the study by Trutschnig [20] that O s corresponds to the topology on C induced by the metric D 2 . So O is finer than O s and O w , i.e., O w O s O .

2.3 Idempotent copulas

For a sub- σ -algebra S of B , a set S S is called an atom in S if it has positive measure and there is no subset E S of S having measure strictly between 0 and λ ( S ) . S is said to be nonatomic if it has no atom; otherwise, it is called atomic. If S has a (countable) family of essentially disjoint atoms B 1 , B 2 , in S for which the sum of their measures is 1, then it is called totally atomic. If S contains only atoms with measure 1, then it is called 1-atomic. For subclasses R and S of B , S is said to be essentially equivalent to R if, for every S S , there exists R R such that λ ( S R ) = 0 .[1] Sub- σ -algebras R and S of B are called essentially equivalent and written as R S if S is essentially equivalent to R and R is essentially equivalent to S .

Recall that an idempotent copula is a copula C such that C 2 = C . It was shown in the studies by Darsow and Olsen [4] and Trutschnig [21] that every idempotent copula is symmetric, which yields that T C L 2 is self-adjoint. So the Markov operator on L 2 of an idempotent copula C is a projection. Clearly, a projection Markov operator is the Markov operator of an idempotent copula. Darsow and Olsen [4] gave a characterization of idempotent copulas C by splitting them into cases according to the size of the σ -algebra γ C of invariant sets under C . Recall that a Borel set S is said to be in γ C if T C 1 S = 1 S . An idempotent copula C is said to be nonatomic/atomic/totally atomic if γ C is nonatomic/atomic/totally atomic, respectively.

Let P = { ( a k , b k ) } k Λ be a partition[2] of I and { C k } k Λ a collection of copulas, where Λ is either N or { 1 , 2 , , m } for some m N . Define P C k , for x , y I by,

P C k ( x , y ) a k + ( b k a k ) C k x a k b k a k , y a k b k a k for x , y ( a k , b k ] , M ( x , y ) , otherwise.

Then, P C k is a copula called the ordinal sum of { C k } with respect to the partition P or the ordinal sum on the partition P with components C k . It is straightforward to verify that ( P C k ) t = P C k t , ( P C k ) ( P D k ) = P ( C k D k ) , and P C k = P D k if and only if C k = D k for all k . Ergo, P C k is idempotent if and only if C k is idempotent for all k . By [4, Lemma 4.1], a Borel subset Q of ( a k , b k ) is invariant under P C k if and only if Q a k b k a k is invariant under C k . A partition { ( a k , b k ) } k Λ of I is called a special partition and written as { ( a k , a k + 1 ) } k Λ if b k = a k + 1 for all k Λ , where a m + 1 b m if Λ = m . A Borel partition of I is a collection { S k } k Λ consisting of essentially pairwise disjoint Borel sets such that λ ( S k ) > 0 for all k and k Λ λ ( S k ) = 1 . Let us quote theorems from the study by Darsow and Olsen [4], some of which have been rewritten to incorporate necessary details from proofs.

Theorem 2.3

Let E be an idempotent copula.

  1. If E is 1-atomic, then E = Π .

  2. If E is nonatomic, then E = C e , f C f , e , where f such that γ E f 1 ( B ) .

Note that the converses of both statements also hold (see [4,19]).

Theorem 2.4

Let P = { ( a k , a k + 1 ) } be a special partition of I and E a copula.

  1. If E = P Π , then E is totally atomic idempotent and γ E σ ( P ) .

  2. Conversely, if P is essentially equivalent to γ E , then

    1. E is an ordinal sum with respect to the partition P ;

    2. if, in addition, γ E σ ( P ) , then E = P E k where each component, E k is 1-atomic copula;

    3. if, in addition to the assumption in (b), E is idempotent, then each E k is the copula Π .

Proposition 2.5

Let { S k } k Λ be a Borel partition of I . Then, there exists p inv and a special partition P = { ( a k , a k + 1 ) } k Λ of I such that p 1 ( S k ) [ a k , a k + 1 ) for all k Λ . Such a p is called a rearrangement of { S k } (into P ).

Theorem 2.6

A copula E is a totally atomic idempotent if and only if E = C p , e ( P Π ) C e , p for some p inv and special partition P of I such that p is a rearrangement of a maximal set of essentially pairwise disjoint atoms { S k } k Λ in γ E into P .

Theorem 2.7

A copula E is an idempotent that is atomic but not totally atomic if and only if E = C p , e ( P F k ) C e , p for some p inv , special partition P of I , and nonatomic idempotent F 1 such that p is a rearrangement of { S k } k Λ into P , where F k = Π for all k 2 , { S k } k 2 is a maximal set of essentially pairwise disjoint atoms in γ E , and S 1 I \ k 2 S k . Note that F 1 = C e , h C h , e for some h .

By Corollaries 3.4.1, 4.4.1, and 5.11 in the study by Darsow and Olsen [4], we have

Corollary 2.8

There is a unique idempotent copula C whose the class γ C of invariant sets is essentially equivalent to a given sub- σ algebra S of B .

2.4 Associated σ -algebras and atomicity of copulas

Following [19], we give some background of associated σ -algebras and various types of the atomicity of copulas. The associated σ -algebras of a copula C or its Markov operator T = T C is defined as follows:

σ C = σ T { S B : T 1 S = 1 R for some R B } and σ C * = σ T * { R B : T 1 S = 1 R for some S B } .

Evidently, if C is idempotent, then σ C = σ C * = γ C . Generally, σ T * = σ T * , and hence σ C * = σ C t . In fact, if T 1 S = 1 R then T * 1 R = 1 S . Consequently, σ C is nonatomic if and only if σ C * is nonatomic. This is also valid for all types of atomicity, i.e., σ C is nonatomic/1-atomic/atomic/totally atomic if and only if σ C * is nonatomic/1-atomic/atomic/totally atomic, respectively. Hence, the atomicity of a copula or its Markov operator is defined corresponding to that of its associated σ -algebras.

Since the associated σ -algebras of a Markov operator are defined via indicator functions which are in L , they all coincide no matter which L p the Markov operator is on, i.e. σ T = σ T L p .

Analogous to the invariant sets, σ A k ’s scale linearly according to P under taking ordinal sum.

Lemma 2.9

If A = P A k where P = { ( a k , b k ) } , then for all Borel sets Q , R ( a k , b k )

  1. Q is in σ A if and only if Q a k b k a k is in σ A k ; and

  2. R is in σ A * if and only if R a k b k a k is in σ A k * .

2.5 Nonatomic copulas

We gather some useful results related to nonatomic copulas below. See [4,16,19].

Proposition 2.10

Let f , g , C C , and S a nonatomic sub- σ -algebra of B .

  1. C f , f = M and C f , g = C f , e C e , g .

  2. C f , e C g , e = C f g , e and C e , g C e , f = C e , f g .

  3. σ C e , f * = σ C f , e f 1 ( B ) .

  4. There exists h such that h 1 ( B ) S .

  5. If C is idempotent and σ C f 1 ( B ) , then C = C e , f C f , e .

  6. σ C e , f C g , e = σ C g , e and σ C e , f C g , e * = σ C e , f * .

  7. σ C f , e C = σ C and σ C C e , f * = σ C * provided that f inv .

  8. σ C C e , f f ( σ C ) and σ C f , e C * f ( σ C * ) provided that f inv .

  9. C e , f C f , e = C e , g C g , e if and only if there is h inv for which f = h g almost everywhere.

  10. If f 1 ( B ) g 1 ( B ) then there exists h inv such that f = h g almost everywhere.

Given p [ 1 , ] and a sub- σ -algebra S of B , we define L p ( S ) to be a subclass of L p consisting only of S -measurable functions.

Theorem 2.11

For every nonatomic copula C, there exist h and g in such that σ C * h 1 ( B ) , σ C g 1 ( B ) , and T C = T e , h T g , e on L 1 ( g 1 ( B ) ) .

A nonatomic copula C is said to be factorizable if there exist f and g in such that C = C e , f C g , e . Given a nonatomic copula C , by Proposition 2.10 (iv), there exist f and g in such that σ C g 1 ( B ) and σ C * f 1 ( B ) . The product C C e , f C f , e C C e , g C g , e , which is independent of the choices of f and g , is called the isoalgebra copula of C . In fact, C is factorizable and shares the same set of associated σ -algebras as that of C . The following theorem presents C as a tool in characterizing factorizable copulas. See [16] for more details.

Theorem 2.12

A nonatomic copula C is factorizable if and only if C coincides with its isoalgebra copula, i.e. C = C .

3 Characterization of pre-idempotent copulas

Definition 3.1

Let C be a copula. The transpose-product of C is defined as C t C , and C is pre-idempotent if its transpose-product is idempotent, i.e., C t C C t C = C t C . Denote { C C : C t C is idempotent } .

Clearly, contains all idempotent copulas as well as all factorizable copulas. In fact, if C = L R , where L and R t are left invertible, then C t C = ( L R ) t ( L R ) = R t R , which is idempotent. By Propositions 2.2 and 2.1,

(3.1) C C t C = C C t C .

Note that there are factorizable copulas, which are not idempotent, such as W , the copula A in Example 3.10, and all asymmetric factorizable copulas.

The following lemma shows that the atomicity of a pre-idempotent copula can be verified via that of its transpose-product.

Lemma 3.2

Let C . Then, σ C = σ C t C and σ C * = σ C C t .

Proof

Let C with corresponding L 2 -Markov operator T C . We will show only that σ C = σ C t C because the other identity can be obtained by substituting C t for C . Let R σ C and S σ C * such that T C 1 R = 1 S . So T C * 1 S = 1 R and T C t C 1 R = T C * ( T C 1 R ) = T C * 1 S = 1 R . Hence, R σ C t C . Conversely, let K σ C t C . Since C t C is idempotent, 1 K = T C t C 1 K = T C * ( T C 1 K ) = T C * g , where g T C 1 K L 2 . By properties of Markov operators, I g d λ = I 1 K d λ = λ ( K ) and 0 g 1 a.e. So 0 g 2 g 1 a.e. Then λ ( K ) = 1 K , 1 K = T C * g , 1 K = g , T C 1 K = g , g = I g 2 d λ I g d λ = λ ( K ) , which implies that g = g 2 a.e. Hence, g = 1 E for some Borel set E . That is, K σ C .□

Let us observe here that if r inv , then C e , r C r , e = M and

(3.2) ( C r , e ( P F k ) C e , p ) t ( C r , e ( P F k ) C e , p ) = C p , e ( P F k t F k ) C e , p .

Proposition 3.3

Let P = { ( a k , a k + 1 ) } k Λ be a special partition of I and p , r inv . The following statements hold for C = C r , e ( P D k ) C e , p , where D k , s are copulas.

  1. If D k = Π for all k , then C is a totally atomic pre-idempotent copula.

  2. If D 1 is factorizable and D k = Π for k 2 , then C is a pre-idempotent copula that is atomic but not totally atomic.

Proof

From equation (3.2), C t C = C p , e ( P D k t D k ) C e , p . If each D k is either factorizable or Π , then each D k t D k , and hence C t C is idempotent, which implies that C is pre-idempotent. The desired atomicity of C in cases 1 and 2 follows from Lemma 3.2 and Theorems 2.4 and 2.7.□

Lemma 3.4

Π is the only 1-atomic pre-idempotent copula.

Proof

Let C be a 1-atomic pre-idempotent. By Lemma 3.2, C t C and C C t are 1-atomic idempotent. So C t C = C C t = Π by Theorem 2.3 (1). To show C = Π , it suffices to show that for each u I , 1 C ( t , u ) = u a.e. t I . Let u I . The derivations

(3.3) u 2 = Π ( u , u ) = C t C ( u , u ) = 0 1 2 C t ( u , t ) 1 C ( t , u ) d t = 0 1 ( 1 C ( t , u ) ) 2 d t and

0 1 ( 1 C ( t , u ) u ) 2 d t = 0 1 ( 1 C ( t , u ) ) 2 d t 0 1 2 u 1 C ( t , u ) d t + 0 1 u 2 d t = 0 1 ( 1 C ( t , u ) ) 2 d t u 2 ,

where the second equality follows from the fact that 0 1 1 C ( t , u ) d t = C ( 1 , u ) C ( 0 , u ) = u , yields 0 1 ( 1 C ( t , u ) u ) 2 d t = 0 , and the claim follows.□

As mentioned above, every factorizable copula is a nonatomic pre-idempotent. The converse also holds.

Theorem 3.5

Every nonatomic pre-idempotent copula is factorizable.

Proof

Let C be a nonatomic pre-idempotent. In light of Theorem 2.11 and Lemma 3.2, there exist f , g such that T C = T e , f T g , e on L 1 ( g 1 ( B ) ) , f 1 ( B ) σ C * = σ C C t , and g 1 ( B ) σ C = σ C t C . By Theorem 2.3(2), the transpose-products of C and C t can be written as C t C = C e , g C g , e and C C t = C e , f C f , e , respectively. By applying equation (3.1) twice, the isoalgebra copula of C equals C C e , f C f , e C C e , g C g , e = C C t C C t C = C C t C = C . By Theorem 2.12, C is factorizable.□

Corollary 3.6

If A is a nonatomic pre-idempotent copula with σ A g 1 ( B ) and σ A * f 1 ( B ) for some f , g , then there exists an invertible copula S such that A = C e , f S C g , e .

Proof

Since the σ -algebras of idempotents A t A and A A t are σ A t A = σ A g 1 ( B ) and σ A A t = σ A * f 1 ( B ) , Theorem 2.3(2) gives A t A = C e , g C g , e and A A t = C e , f C f , e . By Theorem 3.5, A is factorizable: A = C e , h C k , e for some h , k . Then, C e , k C k , e = A t A = C e , g C g , e and C e , h C h , e = A A t = C e , f C f , e . So, by Proposition 2.10(ix), there exist r , p inv such that k = r g and h = p f . Thus, by Proposition 2.10(ii), A = C e , p f C r g , e = C e , f S C g , e , where S C e , p C r , e is invertible.□

The following characterization of factorizable Markov operators is an immediate consequence of Theorem 3.5 and Proposition 2.2.

Corollary 3.7

For every nonatomic Markov operator T on L 2 , T is a partial isometry if and only if T is factorizable.

Theorem 3.8

Let C be atomic, then there exist p , r inv and a special partition P = { ( a k , a k + 1 ) } k Λ such that C = C r , e ( P D k ) C e , p , where D k = Π for all k 2 (if any). Furthermore, D 1 = Π if C is totally atomic; otherwise, D 1 is a factorizable copula.

Proof

By assumption, C t C is atomic idempotent with invariant sets γ C t C = σ C and γ C C t = σ C * . Let us first consider the case that C t C is not totally atomic. Let { S k } k 2 be a maximal collection of essentially pairwise disjoint atoms in σ C , S 1 I \ k 2 S k , and R k ’s Borel sets in σ C * such that T C 1 S k = 1 R k . Consequently, { R k } k 2 is a maximal collection of essentially pairwise disjoint atoms in σ C * . By Proposition 2.5 and Theorem 2.7, there exist a special partition P = { ( a k , a k + 1 ) } of I , a nonatomic idempotent F 1 , and p , r inv such that p 1 ( S k ) [ a k , a k + 1 ) r 1 ( R k ) for all k and C t C = C p , e ( P F k ) C e , p , where F k = Π for k 2 . Put D C e , r C C p , e so that

(3.4) D t D = C e , p C t C C p , e = P F k ,

which is idempotent. For each k Λ , T D 1 [ a k , a k + 1 ) = T e , r T C T p , e 1 p 1 ( S k ) = T e , r T C 1 S k = T e , r 1 R k = 1 r 1 ( R k ) = 1 [ a k , a k + 1 ) . By Theorem 2.4, D is an ordinal sum with respect to the partition P , i.e., D = P D k . So, by equation (3.4), P D k t D k = D t D = P F k . Hence, D 1 t D 1 = F 1 and D k t D k = Π for all k 2 . For k 2 , Lemma 3.4 implies that D k = Π . By Theorem 3.5, the nonatomic pre-idempotent D 1 is factorizable. Thus, C = C r , e D C e , p = C r , e ( P D k ) C e , p , where D k = Π for all k 2 and D 1 is factorizable.

The case that C is totally atomic can be proved in a similar manner as the previous case with minimal adjustments. In this case, { S k } and { R k } are maximal collections of essentially pairwise disjoint atoms in σ C and σ C * , respectively. Theorem 2.6 is applied instead of Theorem 2.7, thus the copulas F 1 and D 1 in this case become the copula Π .□

For conciseness, only in the following theorem, empty intervals will be permissible in the partition P , i.e., the components D k in P D k corresponding to empty intervals vanish. This allows us to unify Lemma 3.4 and Theorems 3.5 and 3.8 and conclude our characterization of pre-idempotent copulas.

Theorem 3.9

For C , there exist invertible measure-preserving functions p and r and a special partition P = { I k } k N of I such that C = C r , e ( P D k ) C e , p , where the component D 1 is factorizable, and each component of D k , k 2 , is the copula Π .

If C is nonatomic, then I 1 = ( 0 , 1 ) and I k = for all k 2 , and the measure-preserving functions r and p can be chosen to be the identity function, hence it is in fact the factorizable copula.

If C is totally atomic, then I 1 = . That is, the ordinal sum P D k contains only the copula Π as its components. In particular, if C is 1-atomic, then I 2 = ( 0 , 1 ) and all other I k s are empty, thus C = Π .

The following example shows that a pre-idempotent with given associated σ -algebras is not necessarily unique.

Example 3.10

  1. Consider a sub- σ -algebra Λ 1 ( B ) , where Λ ( x ) = 2 min { x , 1 x } . Recall [4,19] that D C e , Λ C Λ , e is a nonatomic idempotent copula for which σ D = σ D * Λ 1 ( B ) . Let A C e , Λ C e , w C Λ , e , where w inv is defined as w ( x ) = 1 x . Since w is invertible, it follows from Proposition 2.10 (iii, vi, and vii) that σ A = σ A * Λ 1 ( B ) . The supports of factorizable copulas D and A are shown in Figure 1.

  2. Let a k k 1 3 and P { ( a k , a k + 1 ) , k = 1 , 2 , 3 } be a partition of I . Let F P F k be an ordinal sum on the partition P with components F k Π for all k = 1 , 2 , 3 . Then, F is a totally atomic idempotent copula, hence pre-idempotent, with σ F = σ F * σ ( P ) . Let inv map x to x + 1 3 on [ 1 3 , 2 3 ) , x to x 1 3 on [ 2 3 , 1 ] , and x to itself elsewhere. Then, C e , is an invertible copula. By direct calculation or using results on shuffles of Min (See [17]), B C e , F is a copula whose support is shown in Figure 1. Consider B t B = ( F t C e , t ) ( C e , F ) = F t F = F . Thus, B . By Proposition 2.10 (vii and viii), σ B = σ C e , F = σ F and σ B * = σ C e , F * 1 ( σ F * ) σ F * . But B F .

Figure 1 
               Supports of copulas 
                     
                        
                        
                           D
                        
                        D
                     
                  , 
                     
                        
                        
                           A
                        
                        A
                     
                  , 
                     
                        
                        
                           
                              
                                 C
                              
                              
                                 e
                                 ,
                                 ℓ
                              
                           
                        
                        {C}_{e,\ell }
                     
                  , and 
                     
                        
                        
                           B
                        
                        B
                     
                  .
Figure 1

Supports of copulas D , A , C e , , and B .

Proposition 3.11

Let A , C be such that σ A = σ C and σ A * = σ C * . Then, there exist C 1 , C 2 such that σ C 1 = σ C 1 * = σ C * , σ C 2 * = σ C 2 = σ C , A = C 1 C , and A = C C 2 .

Proof

The case of 1-atomic is clear since A and C are exactly the copula Π by Lemma 3.4.

If A and C are nonatomic sharing the same set of associated σ -algebras, then, by Theorem 3.5, both A and C are factorizable. Let f , g be such that C = C e , f C g , e . Thus, σ A = σ C g 1 ( B ) and σ A * = σ C * f 1 ( B ) . Corollary 3.6 yields h inv such that A = C e , f C e , h C g , e . Then, C e , f C e , h C g , e = ( C e , f C e , h C f , e ) ( C e , f C g , e ) = C 1 C and C e , f C e , h C g , e = ( C e , f C g , e ) ( C e , g C e , h C g , e ) = C C 2 , where C 1 C e , f C e , h C f , e and C 2 C e , g C e , h C g , e are clearly both in . By Proposition 2.10 (ii and vi), σ C 1 = σ C 1 * = σ C * and σ C 2 * = σ C 2 = σ C .

For the atomic case, we first suppose C is atomic but not totally atomic. Let { S k } k 2 and { R k } k 2 be maximal collections of essentially disjoint atoms in σ C and σ C * , respectively, such that T C 1 S k = 1 R k for all k . Then, S 1 I \ k 2 S k and R 1 I \ k 2 R k have positive and equal measure. Since σ A = σ C and σ A * = σ C * , there exists a permutation π on the indices of { S k } k 2 such that T A 1 S k = 1 R π ( k ) . By Theorem 3.8 and its proof, there exist p 1 , p 2 , r 1 , r 2 inv and a special partition P = { ( a k , a k + 1 ) } such that p 1 1 ( S k ) p 2 1 ( S π 1 ( k ) ) [ a k , a k + 1 ) r 2 1 ( R π ( k ) ) r 1 1 ( R k ) , and

(3.5) C = C r 1 , e ( P D k ) C e , p 1 and C r 2 , e ( P E k ) C e , p 1 = A = C r 1 , e ( P F k ) C e , p 2 ,

where D k = E k = F k = Π for all k 2 , and D 1 , E 1 , and F 1 are factorizable copulas with σ D 1 = σ E 1 = σ F 1 and σ D 1 * = σ E 1 * = σ F 1 * . From the nonatomic case, there exist factorizable copulas H 1 and K 1 such that

(3.6) E 1 = H 1 D 1 , F 1 = D 1 K 1 , σ H 1 = σ H 1 * = σ D 1 * , and σ K 1 * = σ K 1 = σ D 1 .

Setting H k = K k = Π for all k 2 , then ( P H k ) ( P D k ) = P H k D k = P E k and ( P D k ) ( P K k ) = P D k K k = P F k , which, together with equation (3.5), gives

A = C r 2 , e ( P E k ) C e , p 1 = C r 2 , e ( P H k ) ( P D k ) C e , p 1 = C r 2 , e ( P H k ) C e , r 1 C = C 1 C and A = C r 1 , e ( P F k ) C e , p 2 = C r 1 , e ( P D k ) ( P K k ) C e , p 2 = C C p 1 , e ( P K k ) C e , p 2 = C C 2 ,

where C 1 = C r 2 , e (