- Email: [email protected]
On the approximation of copulas via shuffles of Min
Statistics and Probability Letters 82 (2012) 1761–1767
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On the approximation of copulas via shuffles of Min Fabrizio Durante a,∗ , Juan Fernández Sánchez b a
School of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy
b
Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain
article
info
Article history: Received 23 April 2012 Received in revised form 10 June 2012 Accepted 10 June 2012 Available online 15 June 2012
abstract We study a multivariate extension of shuffles of Min that has a probabilistic interpretation in terms of mutually completely dependent process. The closure properties of the class of such copulas under different types of convergence is investigated. © 2012 Elsevier B.V. All rights reserved.
MSC: 62H10 62E20 37A05 Keywords: Copula Markov operator Shuffle of Min
1. Introduction For every d ≥ 2, a d-dimensional copula (copula, for brevity) is a distribution function on Id := [0, 1]d whose univariate margins are uniformly distributed. Copulas have been widely studied in statistics since they capture the rank-invariant dependence of continuous random vectors, as shown by Sklar (1959). For more information about copulas and their applications, see, for instance, Nelsen (2006) and Jaworski (2010) and the references therein. Typical investigations about copulas concern the introduction and the study of new (parametric or semi-parametric) families of copulas that can be conveniently fitted to real data. However, one of the main problems of this approach consists of finding a way to point out the family that seems the most convenient for a given situation. In order to avoid such a problem, there has been a growing interest in the approximation of copulas, with the aim to provide some general constructions that could be used to describe a variety of dependence structure. See, for example, some recent literature about Bernstein copulas (Sancetta and Satchell, 2004; Baker, 2008; Mikusiński and Taylor, 2009) and shuffles of copulas (Durante et al., 2009; Durante and Fernández-Sánchez, 2010; Mikusiński and Taylor, 2010; Trutschnig, 2011). A shuffle of Min (where Min indicates the comonotonicity copula Md (u) = min{u1 , . . . , ud }) is any copula obtained by means of a suitable rearrangement of the mass distribution of Md (Mikusiński et al., 1992; Mikusiński and Taylor, 2010). It expresses a special type of dependence, called mutually complete dependence (Lancaster, 1963). Shuffles of Min have been used in several approximation problems, since it has been observed that the class of shuffles of Min is dense in the class of copulas endowed with the L∞ -norm (Mikusiński et al., 1992; Vitale, 1990, 1991; Mikusiński and Taylor, 2010). Recently, different kinds of convergence for copulas have been used as well; in particular M-convergence and ∂ -convergence (for more details, see Mikusiński and Taylor (2010)). Specifically, it has been showed that, under the latter two types of convergence, it is not true that any copula can be approximated by limits of sequences of shuffles of Min.
∗
Corresponding author. E-mail addresses: [email protected] (F. Durante), [email protected] (J.F. Sánchez).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.06.008
1762
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
This fact raises the question about the closure of the class of shuffles of Min with respect to different types of convergence. In Mikusiński and Taylor (2010, Section 7.2) it was conjectured that the closure of the set of shuffles of Min under ∂ -convergence (respectively, M-convergence) coincides with a strict subclass of copulas, which is a possible extension of the shuffles of Min construction to the multivariate case. In this paper, we aim at clarifying some aspects about the notion of shuffles of Min (and shuffles of copulas) that have been introduced in the literature (with particular emphasis on the similarities with the 2-dimensional results by Durante et al. (2009)). In particular, we show that such dependence structures express a kind of functional dependence among the involved random variables (see Section 2). Moreover, we discuss the shuffle approximation with different norms and we prove the conjecture raised by Mikusiński and Taylor (2010) about the closure of shuffles of Min under different norms. 2. Shuffles of copulas in a multivariate setting We recall that a copula C induces a probability measure µC defined, for every non-empty rectangle R ⊆ Id , by µC (R) := VC (R), where VC (R) denotes the C -volume of R, and extended by means of standard arguments to the Borel 1 σ -algebra B (Id ). Such a µC is d-fold stochastic, i.e. µC (p− i (A)) = λ(A) for any Borel set A ⊆ I, where λ is the Lebesgue
measure on I and pi is the i-th projection. Actually, there is a one-to-one correspondence between the class of copulas Cd and the set of d-fold stochastic measures. Following Durante et al. (2009), a suitable definition of shuffles of copulas can be obtained by using some tools from measure theory. Therefore, we recall here the bare minimum that is necessary to understand the concepts we are going to introduce. Let (Ω , F , µ) be a measure space, let (Ω1 , F1 ) be a measurable space and let f : Ω → Ω1 be a measurable function. The push-forward of µ under f is a set function f ∗ µ defined, for every A ∈ F1 , by
(f ∗ µ) (A) = µ f −1 (A) .
(1)
Obviously, the push-forward f ∗ µ is a measure on F1 . Moreover, if µ is a probability, then so is f ∗ µ. A measure-preserving transformation from the measure space (Ω , F , µ) to the measure space (Ω1 , F1 , ν) is a measurable f : Ω → Ω1 such that f ∗ µ = ν , or, equivalently, such that µ ◦ f −1 = ν . We denote by T the set of all measure-preserving transformations of the measure space (I, B (I), λ) and by Tp the set of all T ∈ T such that T is bijective and its inverse T −1 is measure-preserving. By a direct extension of the results by Durante et al. (2009), we give the following definition. Definition 2.1. Let D be a copula with associated measure denoted by µD . A copula C is a shuffle of D if there exist T1 , . . . , Td ∈ Tp such that the measure µC associated with C is given by
µC = (T1 , . . . , Td ) ∗ µD .
(2)
The class of shuffles of D is denoted by S (D). In the bivariate case, the notion of shuffle of D includes the notion of (bivariate) shuffle of a copula introduced in Durante et al. (2009, Definition 7), where T2 = idI , i.e. the identity function of I. Intuitively, a generalized shuffle of D is a way of rearranging the probability mass distribution of D by means of suitable transformations T1 , . . . , Td . Such a rearrangement preserves the constraints about the univariate margins of the copula in view of the fact that every Ti is measure-preserving. In fact, for any Borel set A ⊆ I and for every i ∈ {1, . . . , d},
µC (I × · · · × A × · · · × I) = (T1 , . . . , Td ) ∗ µD (I × · · · × A × · · · × I) = µD (I × · · · × Ti−1 (A) × · · · × I) = λ(Ti−1 (A)) = λ(A). Notice that, if D is an absolutely continuous copula, then every shuffle of D is also absolutely continuous (as can be proved similarly to Durante et al. (2009, Proposition 12)). Remark 2.1. The basic idea of shuffles of copulas goes back to the way how a doubly stochastic matrix can be transformed into another one by means of suitable transformations. In fact, copulas can be consider as a generalization to the infinitedimensional space of such matrices (see, e.g., Kolesárová et al. (2006)). Recently, such a rearrangement of mass has found quite interesting application to the problem of providing sharp bounds for the sums of dependent risks (Puccetti and Rüschendorf, 2012). From a measure-theoretic perspective, generalized shuffles have a quite immediate interpretation, relying on the correspondence between copulas and measure-preserving transformations (see, e.g., Kolesárová et al. (2008) and de Amo et al. (2011)). In fact, we recall that any copula C can be represented in terms of measure-preserving transformation gi : I → I via the expression C (u) = λ g1−1 [0, u1 ] ∩ g2−1 [0, u2 ] ∩ · · · ∩ gd−1 [0, ud ] .
The following result holds.
(3)
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1763
Proposition 2.1. Let C be the copula generated by g1 , . . . , gd as in (3). Then the shuffle of C given by (T1 , . . . , Td ) ∗ µC is a copula generated by T1 ◦ g1 , . . . , Td ◦ gd . Proof. The proof will be carried out for the measure induced by a copula. If C is the copula generated by g1 , . . . , gd , then µC = (g1 , . . . , gd ) ∗ λ. It follows that
(T1 , . . . , Td ) ∗ µC = (T1 , . . . , Td ) ∗ ((g1 , . . . , gd ) ∗ λ) = (T1 ◦ g1 , . . . , Td ◦ gd ) ∗ λ, which is the desired assertion.
Notice that, as stressed by Kolesárová et al. (2008), the representation of copulas in terms of measure-preserving transformations is not unique. In fact, if C is generated by g1 , . . . , gd as in (3), then C is also generated by g1 ◦ ϕ, . . . , gd ◦ ϕ for any measure-preserving bijection ϕ . Now, let us consider in detail the class S (Md ), which contains shuffles of the copula Md . Every C ∈ S (Md ) is called a generalized shuffle of Min. The adjective ‘‘generalized’’ is introduced in analogy to the notation in Durante et al. (2009, Definition 5). In fact, we recall that, in the bivariate case, generalized shuffles of Min extend the notion of shuffle of Min introduced by Mikusiński et al. (1992). Incidentally, S (Md ) coincides with the family of copulas proposed in Mikusiński and Taylor (2010, p. 411). Notice that, if a copula C = (T1 , . . . , Td ) ∗ µMd is a generalized shuffle of Min, then by Proposition 2.1 it follows that C is generated by T1 , . . . , Td . Moreover, since T1 is a bijection, we may express C in an equivalent way in terms of the measurepreserving transformations T1 ◦ T1−1 , T2 ◦ T1−1 , . . . , Td ◦ T1−1 . In other words, C can be written in the form C (u) = λ [0, u1 ] ∩ f2−1 [0, u2 ] ∩ · · · ∩ fd−1 [0, ud ]
(4)
for suitable λ-measure-preserving fi ∈ Tp . The following characterization holds. Proposition 2.2. Let C be a copula. The following statements are equivalent: (a) µC = (T1 , . . . , Td ) ∗ µMd for some T1 , . . . , Td ∈ Tp ; (b) C concentrates the probability mass on the set {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))} for suitable fi ∈ Tp . (c) there exists a random vector U = (U1 , . . . , Ud ) on the probability space (Ω , F , P) having distribution function equal to C for which it holds that, for every i, j ∈ {1, . . . , d}, i ̸= j, there exists gij ∈ Tp such that Ui = gij (Uj ) almost surely. Proof. (a) H⇒ (b): Let T = (T1 , . . . , Td ). Since µMd concentrates the probability mass on the set Γ = {u ∈ Id : u1 = u2 = · · · = ud }, we have
µC (T(Γ )) = T ∗ µMd (T(Γ )) = µMd (Γ ) = 1, from which it follows that C concentrates the probability mass on T(Γ ) = {u = (T1 (t ), . . . , Td (t )), t ∈ I}
= {u = (s, T2 (T1−1 (s)), . . . , Td (T1−1 (s))), s ∈ I}, which is the desired assertion. (b) H⇒ (c): Let U be a random vector distributed according to the copula C . If C satisfies (b), then
P U ∈ {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))} = 1,
from which it follows that each component of U is almost surely a function of the other via a transformation in Tp . (c) H⇒ (a): Because of (c), if the random vector U is distributed according to the copula C , then it is almost surely equal to the vector (U1 , f2 (U1 ), . . . , fd (U1 )) for suitable f2 , . . . , fd ∈ Tp . It follows that
µC = (U1 , f2 (U1 ), . . . , fd (U1 )) ∗ P = ((idI , f2 , . . . , fd ) ◦ (U1 , U1 , . . . , U1 )) ∗ P = (idI , f2 , . . . , fd ) ∗ µM , which implies (a).
Property (c) of Proposition 2.2 is equivalent to the fact that all the bivariate random pairs extracted from (U1 , . . . , Ud ) are generalized shuffles of Min in the sense of Durante et al. (2009). Specifically, any pair (Ui , Uj ) is mutually complete dependent in the sense of Lancaster (1963). This latter characterization is, in some sense, an extension of the comonotonicity property of a random vector (see, e.g., Dhaene et al. (2002) and Puccetti and Scarsini (2010) and the references therein). In fact, while comonotonicity of a vector is equivalent to say that each of its components is a monotone increasing function of the others, mutually complete dependence implies that each component of a vector is a bijective transformation of the others.
1764
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
Remark 2.2. Notice that, if C ∈ S (Md ), then the first derivatives of C take values on {0, 1}. This can be shown as a consequence of Proposition 2.2 (part b) by using similar arguments as in Darsow and Olsen (2010). The multivariate shuffles of Min discussed in Mikusiński and Taylor (2010) can be considered as a subclass of S (Md ) (as noted by Durante et al. (2009) for the bivariate case). This assertion is made clear in the following result. Here, we assume that a function is piece-wise continuous if it is defined on a non-degenerate interval and has at most finitely many discontinuities, all of them being jumps. Corollary 2.3. Let C be a multivariate shuffle of Min in the sense of Mikusiński and Taylor (2010). The following statements are equivalent: (a) C = (T1 , . . . , Td ) ∗ µMd for suitable piece-wise continuous Ti ∈ Tp . (b) C concentrates the probability mass on the set {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))}, for suitable piece-wise continuous fi ∈ Tp . (c) there exists a random vector (U1 , . . . , Ud ) having distribution function equal to C for which it holds that, for every i, j ∈ {1, . . . , d}, i ̸= j, there exists a piece-wise continuous gij ∈ Tp such that Ui = gij (Uj ) almost surely. In other words, while generalized shuffles of Min are generated by means of bijective measure-preserving transformations, shuffles of Min are generated by those bijections that are also piece-wise continuous. Obviously, one notion is strictly contained in the other. 3. Approximation under different types of convergence Shuffles of Min play a role in several studies about approximation of copulas. In particular, as known, in the topology induced by uniform convergence, multivariate shuffles of Min are dense in Cd (see, for instance, Mikusiński and Taylor (2010) and also Vitale (1991)). However, such a result cannot be generalized to other kinds of convergence, like for instance ∂ -convergence defined below. This poses the question about the closure of the class of shuffles of Min under different topologies. For instance, it has been showed that, in the bivariate case, the closure of shuffles of Min with respect to Sobolev norm is given by the class of mutually completely dependent copulas (i.e. generalized shuffles of copulas): see for instance, Siburg and Stoimenov (2010) and Trutschnig (2011). A result that has been used in order to define measures of mutual complete dependence. Here we are interested in some approximation results defined for the following types of convergence introduced by Mikusiński and Taylor (2010) (we refer to the original paper for more details about them). ∂
Definition 3.1. Let (Cn )n and C be copulas. We say that Cn converges to C (as n → ∞) in the ∂ -convergence (write: Cn → C ) if, for every i ∈ {1, . . . , d},
∂ Cn ∂C ( x , . . . , x , t , . . . , x ) − ( x , . . . , x , t , . . . , x ) 1 i −1 d 1 i−1 d dt = 0. n→+∞ I ∂ xi ∂ xi lim
(5)
Following Mikusiński and Taylor (2009), we recall that the Markov operator T associated with C with respect to the i-th component is uniquely determined by the equality 1
Th(x)g (x)dx = 0
Id
g (xi )h(x1 , . . . , xi−1 , xi+1 , . . . , xd )dµC (x1 , x2 , . . . , xd )
for all h ∈ L∞ (Id−1 , µ′ ) and g ∈ L∞ (I, λ), where µ′ is the probability measure associated with the (d − 1)-marginal of C with respect to the i-th component. M
Definition 3.2. Let (Cn )n and C be copulas. We say that Cn converges to C (as n → ∞) in the M-convergence (write: Cn → C ) if, for all k ∈ {1, 2, . . . , d} and for all a.e. bounded and measurable h: Id−1 → R we have
lim
n→∞
1
|Tnk h(t ) − T k h(t )|dt = 0
0
where T k is the Markov operator associated with µC with respect to the k-th component, and Tnk is the Markov operator associated with µCn with respect to the k-th component (for these definitions, see Mikusiński and Taylor (2009)). From Mikusiński and Taylor (2010, Theorem 10) it follows that M-convergence implies ∂ -convergence. The closure of the class of shuffles of Min under the topology induced by these kinds of convergences has been posed by Mikusiński and Taylor (2010). Here we provide an answer. Proposition 3.1. The class S (Md ) of generalized shuffles of Min is: (a) the closure of the class of shuffles of Min in the topology induced by the ∂ -convergence on Cd ; (b) the closure of the class of shuffles of Min in the topology induced by the M-convergence on Cd .
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1765
The proof of the above results needs a series of preliminary lemmas. Lemma 3.2. Let f be a measure-preserving function over I. Then there exists a sequence of one-to-one piecewise linear (with slope 1 or −1) measure-preserving functions over I converging to f almost everywhere. Proof. The proof is contained in Chou and Nguyen (1990, Theorem 2.2).
Lemma 3.3 (Disintegration Theorem). Let µ be a probability measure on Id . Let π : Id → I be a Borel function and let ν be the push-forward of µ under π . Then there exists a ν -a.e. uniquely determined Borel family of probability measures {µx }x∈I on Id such that
µx (Id \ π −1 (x)) = 0 for ν -a.e. x ∈ I.
(6)
and
Id
h(x)dµ(x) =
π −1 (x)
I
h(x)dµx (x) dν(x),
(7)
for every Borel map h: Id → [0, +∞]. Proof. The proof is contained, for instance, in Ambrosio et al. (2008, Theorem 5.3.1).
Lemma 3.4. Suppose that a sequence (Cn )n of shuffles of Min tends in the sense of ∂ -convergence to a copula C , as n goes to ∞. Then C ∈ S (Md ). Proof. Suppose that a sequence (Cn )n of shuffles of Min tends in the ∂ -convergence to a copula C as n → ∞. Then, for every x ∈ Id and for every i ∈ {1, . . . , d} it follows that L1
∂i Cn (x1 , . . . , xi−1 , t , . . . , xd ) −→ ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) as n tends to ∞. Therefore, there exists a subsequence (Cσ (n) )n ⊆ (Cn )n such that ∂i Cσ (n) (x1 , . . . , xi−1 , t , . . . , xd ) tends to ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) pointwise (as n → ∞) for almost all t ∈ I. Since every Cσ (n) is a shuffle of Min, one has
∂i Cσ (n) (x1 , . . . , xi−1 , t , . . . , xd ) ∈ {0, 1} almost everywhere, which implies that ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) ∈ {0, 1} almost everywhere. Let x2 , . . . , xd ∈ I. In view of the Lemma 3.3 C ( x1 , . . . , xd ) =
x1
[0,x2 ]×···×[0,xd ]
0
dµx (x) dx.
Moreover, since C is absolutely continuous in each argument, it follows that C ( x1 , . . . , xd ) =
x1
∂1 C (x, x2 , . . . , xd )dx,
0
which implies that for almost all x ∈ I
µx ([0, x2 ] × · · · × [0, xd ]) =
[0,x2 ]×···×[0,xd ]
dµx (x)
= ∂1 C (x, x2 , . . . , xd ) ∈ {0, 1} . In view of Fubini Theorem it follows that in a subset of Id of measure 1 we have
µx ([0, x2 ] × · · · × [0, xd ]) ∈ {0, 1} . Therefore, µx is a Dirac measure for almost all x ∈ I and it concentrates the probability mass in a graph of a function of type (f2 , . . . , fd ). By repeating the above procedure along each of the axis, we may find that each fi is bijective. Thus, in view of Proposition 2.2(part b), C is a generalized shuffle of Min. Lemma 3.5. Let C ∈ S (Md ). Then there exists a a sequence (Cn )n of shuffles of Min that tends (in the M-convergence) to C , as n tends to ∞. Proof. Let C ∈ S (Md ). The Markov operator T associated with C is uniquely determined by the equality 1
Th(x)g (x)dx = 0
Id
g (x)h(x2 , . . . , xd )dµC (x, x2 , . . . , xd )
1766
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
for all h ∈ L∞ (Id−1 , µ′ ), µ′ (S ) = µC (I × S ) for any Borel set S ⊆ Id−1 , and g ∈ L∞ (I, λ). By applying Lemma 3.3 to the right hand side of the previous equality we get
Id
g (x)h(x2 , . . . , xd )dµC (x, x2 , . . . , xd ) =
1
g ( x)
Id−1
0
h(x2 , . . . , xd )dµx (x2 , . . . , xd ) dx.
It follows that for almost all x ∈ I Th(x) =
Id−1
h(x2 , . . . , xd )dµx (x2 , . . . , xd ).
Since C concentrates the mass on the set {(x, f2 (x), . . . , fd (x)): x ∈ I}, it follows that Th(x) = h(f2 (x), . . . , fd (x)) M
for almost all x ∈ I. Now, it is not difficult to see that Cn → C if, and only if, 1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx → 0, 0
where f2,n (x), . . . , fd,n are the measure-preserving transformations that support the graph where the mass of Cn is concentrated. In view of Lemma 3.2, for i ∈ {2, . . . , d} any fi can be approximated in the pointwise convergence by a sequence of measure-preserving transformations si,n that are piecewise linear. Since these functions are bounded, it can be derived from the Lebesgue Theorem that the convergence is guaranteed in the L1 -sense. Now, let h be a continuous function on I. Since h is also uniformly continuous on I, it follows that for ε > 0, there exists η > 0 such that, if maxi=2,...,n |xi − yi | < η, then |h(x2 , . . . , xd ) − h(y2 , . . . , yd )| < ε/2. Let η1 < min{η, ε/2, ε/(2d∥h∥)}. We may choose n ∈ N in such a way that 1
|fi (x) − si,n (x)|dx < η12 . 0
This implies that |fi (x) − si,n (x)| < η1 in a set Ai of measure 1 − η1 . Let A = ∩di=2 Ai and let B = I \ A. Then we have
1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx ≤ |h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx + ∥h∥dx < ε.
0
A
Since continuous functions are dense in the class of L
B
∞
functions it follows that
1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx → 0, 0
for all h ∈ L∞ (Id−1 , µ′ ), which is the desired assertion.
Proof of Proposition 3.1. (a) From Lemma 3.4 it follows that any sequence of shuffles of Min ∂ -converges to an element of S (Md ). Moreover, in view of Lemma 3.5 and the fact that M-convergence implies ∂ -convergence, any element of S (Md ) is a limit of shuffles of Min in the ∂ -convergence. This concludes the proof. (b) Analogously, from Lemma 3.4 and and the fact that M-convergence implies ∂ -convergence, it follows that any sequence of shuffles of Min M-converges to an element C that must be in S (Md ). Moreover, in view of Lemma 3.5, any element of S (Md ) is a a limit of shuffles of Min in the M-convergence. This concludes the proof. Remark 3.1. An example of a copula which is a generalized shuffle of Min, but not a shuffle of Min, can be given by the method described in Trutschnig (2012, Example 1). Specifically, we take a sequence of shuffles of Min by considering the probability mass of M2 in each of the squares described in Trutschnig (2012, Fig. 1). Then the limiting copula (in any of the above introduced convergences) is a generalized shuffle of Min. Acknowledgments We acknowledge the Reviewers for the careful reading and useful suggestions that have improved the manuscript. The first author acknowledges the support of the Free University of Bozen-Bolzano, School of Economics and Management, via the project ‘‘Risk and Dependence’’. The first and second author have been supported by the Ministerio de Ciencia e Innovación (Spain) under research project MTM2011-22394.
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1767
References Ambrosio, L., Gigli, N., Savaré, G., 2008. Gradient Flows in Metric Spaces and in the Space of Probability Measures, second ed. In: Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. Baker, R., 2008. An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99 (10), 2312–2327. Chou, S.-H., Nguyen, T.T., 1990. On Fréchet theorem in the set of measure preserving functions over the unit interval. Internat. J. Math. Math. Sci. 13 (2), 373–378. Darsow, W.F., Olsen, E.T., 2010. Characterization of idempotent 2-copulas. Note Mat. 30 (1), 193–235. de Amo, E., Díaz-Carrillo, M., Fernández-Sánchez, J., 2011. Measure-preserving functions and the independence copula. Mediterr. J. Math. 8 (3), 431–451. Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3–33. Durante, F., Fernández-Sánchez, J., 2010. Multivariate shuffles and approximation of copulas. Statist. Probab. Lett. 80 (23–24), 1827–1834. Durante, F., Sarkoci, P., Sempi, C., 2009. Shuffles of copulas. J. Math. Anal. Appl. 352 (2), 914–921. Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (Eds.), 2010. Copula Theory and its Applications. In: Lecture Notes in Statistics—Proceedings, vol. 198. Springer, Berlin, Heidelberg. Kolesárová, A., Mesiar, R., Mordelová, J., Sempi, C., 2006. Discrete copulas. IEEE Trans. Fuzzy Syst. 14 (5), 698–705. Kolesárová, A., Mesiar, R., Sempi, C., 2008. Measure-preserving transformations, copulæand compatibility. Mediterr. J. Math. 5 (3), 325–339. Lancaster, H.O., 1963. Correlation and complete dependence of random variables. Ann. Math. Statist. 34, 1315–1321. Mikusiński, P., Sherwood, H., Taylor, M.D., 1992. Shuffles of Min. Stochastica 13 (1), 61–74. Mikusiński, P., Taylor, M.D., 2009. Markov operators and n-copulas. Ann. Polon. Math. 96 (1), 75–95. Mikusiński, P., Taylor, M.D., 2010. Some approximations of n-copulas. Metrika 72 (3), 385–414. Nelsen, R.B., 2006. An Introduction to Copulas, second ed. In: Springer Series in Statistics, Springer, New York. Puccetti, G., Rüschendorf, L., 2012. Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236 (7), 1833–1840. Puccetti, G., Scarsini, M., 2010. Multivariate comonotonicity. J. Multivariate Anal. 101 (1), 291–304. Sancetta, A., Satchell, S., 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20 (3), 535–562. Siburg, K.F., Stoimenov, P.A., 2010. A measure of mutual complete dependence. Metrika 71 (2), 239–251. Sklar, A., 1959. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231. Trutschnig, W., 2011. On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384 (2), 690–705. Trutschnig, W., 2012. Some results on the convergence of (quasi-)copulas. Fuzzy Sets and Systems 191, 113–121. Vitale, R.A., 1990. On stochastic dependence and a class of degenerate distributions. In: Topics in Statistical Dependence (Somerset, PA, 1987). In: IMS Lecture Notes Monogr. Ser. Inst. Math. Statist., vol. 16, Hayward, CA, pp. 459–469. Vitale, R.A., 1991. Approximation by mutually completely dependent processes. J. Approx. Theory 66 (2), 225–228.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On the approximation of copulas via shuffles of Min Fabrizio Durante a,∗ , Juan Fernández Sánchez b a
School of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy
b
Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain
article
info
Article history: Received 23 April 2012 Received in revised form 10 June 2012 Accepted 10 June 2012 Available online 15 June 2012
abstract We study a multivariate extension of shuffles of Min that has a probabilistic interpretation in terms of mutually completely dependent process. The closure properties of the class of such copulas under different types of convergence is investigated. © 2012 Elsevier B.V. All rights reserved.
MSC: 62H10 62E20 37A05 Keywords: Copula Markov operator Shuffle of Min
1. Introduction For every d ≥ 2, a d-dimensional copula (copula, for brevity) is a distribution function on Id := [0, 1]d whose univariate margins are uniformly distributed. Copulas have been widely studied in statistics since they capture the rank-invariant dependence of continuous random vectors, as shown by Sklar (1959). For more information about copulas and their applications, see, for instance, Nelsen (2006) and Jaworski (2010) and the references therein. Typical investigations about copulas concern the introduction and the study of new (parametric or semi-parametric) families of copulas that can be conveniently fitted to real data. However, one of the main problems of this approach consists of finding a way to point out the family that seems the most convenient for a given situation. In order to avoid such a problem, there has been a growing interest in the approximation of copulas, with the aim to provide some general constructions that could be used to describe a variety of dependence structure. See, for example, some recent literature about Bernstein copulas (Sancetta and Satchell, 2004; Baker, 2008; Mikusiński and Taylor, 2009) and shuffles of copulas (Durante et al., 2009; Durante and Fernández-Sánchez, 2010; Mikusiński and Taylor, 2010; Trutschnig, 2011). A shuffle of Min (where Min indicates the comonotonicity copula Md (u) = min{u1 , . . . , ud }) is any copula obtained by means of a suitable rearrangement of the mass distribution of Md (Mikusiński et al., 1992; Mikusiński and Taylor, 2010). It expresses a special type of dependence, called mutually complete dependence (Lancaster, 1963). Shuffles of Min have been used in several approximation problems, since it has been observed that the class of shuffles of Min is dense in the class of copulas endowed with the L∞ -norm (Mikusiński et al., 1992; Vitale, 1990, 1991; Mikusiński and Taylor, 2010). Recently, different kinds of convergence for copulas have been used as well; in particular M-convergence and ∂ -convergence (for more details, see Mikusiński and Taylor (2010)). Specifically, it has been showed that, under the latter two types of convergence, it is not true that any copula can be approximated by limits of sequences of shuffles of Min.
∗
Corresponding author. E-mail addresses: [email protected] (F. Durante), [email protected] (J.F. Sánchez).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.06.008
1762
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
This fact raises the question about the closure of the class of shuffles of Min with respect to different types of convergence. In Mikusiński and Taylor (2010, Section 7.2) it was conjectured that the closure of the set of shuffles of Min under ∂ -convergence (respectively, M-convergence) coincides with a strict subclass of copulas, which is a possible extension of the shuffles of Min construction to the multivariate case. In this paper, we aim at clarifying some aspects about the notion of shuffles of Min (and shuffles of copulas) that have been introduced in the literature (with particular emphasis on the similarities with the 2-dimensional results by Durante et al. (2009)). In particular, we show that such dependence structures express a kind of functional dependence among the involved random variables (see Section 2). Moreover, we discuss the shuffle approximation with different norms and we prove the conjecture raised by Mikusiński and Taylor (2010) about the closure of shuffles of Min under different norms. 2. Shuffles of copulas in a multivariate setting We recall that a copula C induces a probability measure µC defined, for every non-empty rectangle R ⊆ Id , by µC (R) := VC (R), where VC (R) denotes the C -volume of R, and extended by means of standard arguments to the Borel 1 σ -algebra B (Id ). Such a µC is d-fold stochastic, i.e. µC (p− i (A)) = λ(A) for any Borel set A ⊆ I, where λ is the Lebesgue
measure on I and pi is the i-th projection. Actually, there is a one-to-one correspondence between the class of copulas Cd and the set of d-fold stochastic measures. Following Durante et al. (2009), a suitable definition of shuffles of copulas can be obtained by using some tools from measure theory. Therefore, we recall here the bare minimum that is necessary to understand the concepts we are going to introduce. Let (Ω , F , µ) be a measure space, let (Ω1 , F1 ) be a measurable space and let f : Ω → Ω1 be a measurable function. The push-forward of µ under f is a set function f ∗ µ defined, for every A ∈ F1 , by
(f ∗ µ) (A) = µ f −1 (A) .
(1)
Obviously, the push-forward f ∗ µ is a measure on F1 . Moreover, if µ is a probability, then so is f ∗ µ. A measure-preserving transformation from the measure space (Ω , F , µ) to the measure space (Ω1 , F1 , ν) is a measurable f : Ω → Ω1 such that f ∗ µ = ν , or, equivalently, such that µ ◦ f −1 = ν . We denote by T the set of all measure-preserving transformations of the measure space (I, B (I), λ) and by Tp the set of all T ∈ T such that T is bijective and its inverse T −1 is measure-preserving. By a direct extension of the results by Durante et al. (2009), we give the following definition. Definition 2.1. Let D be a copula with associated measure denoted by µD . A copula C is a shuffle of D if there exist T1 , . . . , Td ∈ Tp such that the measure µC associated with C is given by
µC = (T1 , . . . , Td ) ∗ µD .
(2)
The class of shuffles of D is denoted by S (D). In the bivariate case, the notion of shuffle of D includes the notion of (bivariate) shuffle of a copula introduced in Durante et al. (2009, Definition 7), where T2 = idI , i.e. the identity function of I. Intuitively, a generalized shuffle of D is a way of rearranging the probability mass distribution of D by means of suitable transformations T1 , . . . , Td . Such a rearrangement preserves the constraints about the univariate margins of the copula in view of the fact that every Ti is measure-preserving. In fact, for any Borel set A ⊆ I and for every i ∈ {1, . . . , d},
µC (I × · · · × A × · · · × I) = (T1 , . . . , Td ) ∗ µD (I × · · · × A × · · · × I) = µD (I × · · · × Ti−1 (A) × · · · × I) = λ(Ti−1 (A)) = λ(A). Notice that, if D is an absolutely continuous copula, then every shuffle of D is also absolutely continuous (as can be proved similarly to Durante et al. (2009, Proposition 12)). Remark 2.1. The basic idea of shuffles of copulas goes back to the way how a doubly stochastic matrix can be transformed into another one by means of suitable transformations. In fact, copulas can be consider as a generalization to the infinitedimensional space of such matrices (see, e.g., Kolesárová et al. (2006)). Recently, such a rearrangement of mass has found quite interesting application to the problem of providing sharp bounds for the sums of dependent risks (Puccetti and Rüschendorf, 2012). From a measure-theoretic perspective, generalized shuffles have a quite immediate interpretation, relying on the correspondence between copulas and measure-preserving transformations (see, e.g., Kolesárová et al. (2008) and de Amo et al. (2011)). In fact, we recall that any copula C can be represented in terms of measure-preserving transformation gi : I → I via the expression C (u) = λ g1−1 [0, u1 ] ∩ g2−1 [0, u2 ] ∩ · · · ∩ gd−1 [0, ud ] .
The following result holds.
(3)
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1763
Proposition 2.1. Let C be the copula generated by g1 , . . . , gd as in (3). Then the shuffle of C given by (T1 , . . . , Td ) ∗ µC is a copula generated by T1 ◦ g1 , . . . , Td ◦ gd . Proof. The proof will be carried out for the measure induced by a copula. If C is the copula generated by g1 , . . . , gd , then µC = (g1 , . . . , gd ) ∗ λ. It follows that
(T1 , . . . , Td ) ∗ µC = (T1 , . . . , Td ) ∗ ((g1 , . . . , gd ) ∗ λ) = (T1 ◦ g1 , . . . , Td ◦ gd ) ∗ λ, which is the desired assertion.
Notice that, as stressed by Kolesárová et al. (2008), the representation of copulas in terms of measure-preserving transformations is not unique. In fact, if C is generated by g1 , . . . , gd as in (3), then C is also generated by g1 ◦ ϕ, . . . , gd ◦ ϕ for any measure-preserving bijection ϕ . Now, let us consider in detail the class S (Md ), which contains shuffles of the copula Md . Every C ∈ S (Md ) is called a generalized shuffle of Min. The adjective ‘‘generalized’’ is introduced in analogy to the notation in Durante et al. (2009, Definition 5). In fact, we recall that, in the bivariate case, generalized shuffles of Min extend the notion of shuffle of Min introduced by Mikusiński et al. (1992). Incidentally, S (Md ) coincides with the family of copulas proposed in Mikusiński and Taylor (2010, p. 411). Notice that, if a copula C = (T1 , . . . , Td ) ∗ µMd is a generalized shuffle of Min, then by Proposition 2.1 it follows that C is generated by T1 , . . . , Td . Moreover, since T1 is a bijection, we may express C in an equivalent way in terms of the measurepreserving transformations T1 ◦ T1−1 , T2 ◦ T1−1 , . . . , Td ◦ T1−1 . In other words, C can be written in the form C (u) = λ [0, u1 ] ∩ f2−1 [0, u2 ] ∩ · · · ∩ fd−1 [0, ud ]
(4)
for suitable λ-measure-preserving fi ∈ Tp . The following characterization holds. Proposition 2.2. Let C be a copula. The following statements are equivalent: (a) µC = (T1 , . . . , Td ) ∗ µMd for some T1 , . . . , Td ∈ Tp ; (b) C concentrates the probability mass on the set {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))} for suitable fi ∈ Tp . (c) there exists a random vector U = (U1 , . . . , Ud ) on the probability space (Ω , F , P) having distribution function equal to C for which it holds that, for every i, j ∈ {1, . . . , d}, i ̸= j, there exists gij ∈ Tp such that Ui = gij (Uj ) almost surely. Proof. (a) H⇒ (b): Let T = (T1 , . . . , Td ). Since µMd concentrates the probability mass on the set Γ = {u ∈ Id : u1 = u2 = · · · = ud }, we have
µC (T(Γ )) = T ∗ µMd (T(Γ )) = µMd (Γ ) = 1, from which it follows that C concentrates the probability mass on T(Γ ) = {u = (T1 (t ), . . . , Td (t )), t ∈ I}
= {u = (s, T2 (T1−1 (s)), . . . , Td (T1−1 (s))), s ∈ I}, which is the desired assertion. (b) H⇒ (c): Let U be a random vector distributed according to the copula C . If C satisfies (b), then
P U ∈ {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))} = 1,
from which it follows that each component of U is almost surely a function of the other via a transformation in Tp . (c) H⇒ (a): Because of (c), if the random vector U is distributed according to the copula C , then it is almost surely equal to the vector (U1 , f2 (U1 ), . . . , fd (U1 )) for suitable f2 , . . . , fd ∈ Tp . It follows that
µC = (U1 , f2 (U1 ), . . . , fd (U1 )) ∗ P = ((idI , f2 , . . . , fd ) ◦ (U1 , U1 , . . . , U1 )) ∗ P = (idI , f2 , . . . , fd ) ∗ µM , which implies (a).
Property (c) of Proposition 2.2 is equivalent to the fact that all the bivariate random pairs extracted from (U1 , . . . , Ud ) are generalized shuffles of Min in the sense of Durante et al. (2009). Specifically, any pair (Ui , Uj ) is mutually complete dependent in the sense of Lancaster (1963). This latter characterization is, in some sense, an extension of the comonotonicity property of a random vector (see, e.g., Dhaene et al. (2002) and Puccetti and Scarsini (2010) and the references therein). In fact, while comonotonicity of a vector is equivalent to say that each of its components is a monotone increasing function of the others, mutually complete dependence implies that each component of a vector is a bijective transformation of the others.
1764
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
Remark 2.2. Notice that, if C ∈ S (Md ), then the first derivatives of C take values on {0, 1}. This can be shown as a consequence of Proposition 2.2 (part b) by using similar arguments as in Darsow and Olsen (2010). The multivariate shuffles of Min discussed in Mikusiński and Taylor (2010) can be considered as a subclass of S (Md ) (as noted by Durante et al. (2009) for the bivariate case). This assertion is made clear in the following result. Here, we assume that a function is piece-wise continuous if it is defined on a non-degenerate interval and has at most finitely many discontinuities, all of them being jumps. Corollary 2.3. Let C be a multivariate shuffle of Min in the sense of Mikusiński and Taylor (2010). The following statements are equivalent: (a) C = (T1 , . . . , Td ) ∗ µMd for suitable piece-wise continuous Ti ∈ Tp . (b) C concentrates the probability mass on the set {x ∈ Id | x = (x1 , f2 (x1 ), . . . , fd (x1 ))}, for suitable piece-wise continuous fi ∈ Tp . (c) there exists a random vector (U1 , . . . , Ud ) having distribution function equal to C for which it holds that, for every i, j ∈ {1, . . . , d}, i ̸= j, there exists a piece-wise continuous gij ∈ Tp such that Ui = gij (Uj ) almost surely. In other words, while generalized shuffles of Min are generated by means of bijective measure-preserving transformations, shuffles of Min are generated by those bijections that are also piece-wise continuous. Obviously, one notion is strictly contained in the other. 3. Approximation under different types of convergence Shuffles of Min play a role in several studies about approximation of copulas. In particular, as known, in the topology induced by uniform convergence, multivariate shuffles of Min are dense in Cd (see, for instance, Mikusiński and Taylor (2010) and also Vitale (1991)). However, such a result cannot be generalized to other kinds of convergence, like for instance ∂ -convergence defined below. This poses the question about the closure of the class of shuffles of Min under different topologies. For instance, it has been showed that, in the bivariate case, the closure of shuffles of Min with respect to Sobolev norm is given by the class of mutually completely dependent copulas (i.e. generalized shuffles of copulas): see for instance, Siburg and Stoimenov (2010) and Trutschnig (2011). A result that has been used in order to define measures of mutual complete dependence. Here we are interested in some approximation results defined for the following types of convergence introduced by Mikusiński and Taylor (2010) (we refer to the original paper for more details about them). ∂
Definition 3.1. Let (Cn )n and C be copulas. We say that Cn converges to C (as n → ∞) in the ∂ -convergence (write: Cn → C ) if, for every i ∈ {1, . . . , d},
∂ Cn ∂C ( x , . . . , x , t , . . . , x ) − ( x , . . . , x , t , . . . , x ) 1 i −1 d 1 i−1 d dt = 0. n→+∞ I ∂ xi ∂ xi lim
(5)
Following Mikusiński and Taylor (2009), we recall that the Markov operator T associated with C with respect to the i-th component is uniquely determined by the equality 1
Th(x)g (x)dx = 0
Id
g (xi )h(x1 , . . . , xi−1 , xi+1 , . . . , xd )dµC (x1 , x2 , . . . , xd )
for all h ∈ L∞ (Id−1 , µ′ ) and g ∈ L∞ (I, λ), where µ′ is the probability measure associated with the (d − 1)-marginal of C with respect to the i-th component. M
Definition 3.2. Let (Cn )n and C be copulas. We say that Cn converges to C (as n → ∞) in the M-convergence (write: Cn → C ) if, for all k ∈ {1, 2, . . . , d} and for all a.e. bounded and measurable h: Id−1 → R we have
lim
n→∞
1
|Tnk h(t ) − T k h(t )|dt = 0
0
where T k is the Markov operator associated with µC with respect to the k-th component, and Tnk is the Markov operator associated with µCn with respect to the k-th component (for these definitions, see Mikusiński and Taylor (2009)). From Mikusiński and Taylor (2010, Theorem 10) it follows that M-convergence implies ∂ -convergence. The closure of the class of shuffles of Min under the topology induced by these kinds of convergences has been posed by Mikusiński and Taylor (2010). Here we provide an answer. Proposition 3.1. The class S (Md ) of generalized shuffles of Min is: (a) the closure of the class of shuffles of Min in the topology induced by the ∂ -convergence on Cd ; (b) the closure of the class of shuffles of Min in the topology induced by the M-convergence on Cd .
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1765
The proof of the above results needs a series of preliminary lemmas. Lemma 3.2. Let f be a measure-preserving function over I. Then there exists a sequence of one-to-one piecewise linear (with slope 1 or −1) measure-preserving functions over I converging to f almost everywhere. Proof. The proof is contained in Chou and Nguyen (1990, Theorem 2.2).
Lemma 3.3 (Disintegration Theorem). Let µ be a probability measure on Id . Let π : Id → I be a Borel function and let ν be the push-forward of µ under π . Then there exists a ν -a.e. uniquely determined Borel family of probability measures {µx }x∈I on Id such that
µx (Id \ π −1 (x)) = 0 for ν -a.e. x ∈ I.
(6)
and
Id
h(x)dµ(x) =
π −1 (x)
I
h(x)dµx (x) dν(x),
(7)
for every Borel map h: Id → [0, +∞]. Proof. The proof is contained, for instance, in Ambrosio et al. (2008, Theorem 5.3.1).
Lemma 3.4. Suppose that a sequence (Cn )n of shuffles of Min tends in the sense of ∂ -convergence to a copula C , as n goes to ∞. Then C ∈ S (Md ). Proof. Suppose that a sequence (Cn )n of shuffles of Min tends in the ∂ -convergence to a copula C as n → ∞. Then, for every x ∈ Id and for every i ∈ {1, . . . , d} it follows that L1
∂i Cn (x1 , . . . , xi−1 , t , . . . , xd ) −→ ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) as n tends to ∞. Therefore, there exists a subsequence (Cσ (n) )n ⊆ (Cn )n such that ∂i Cσ (n) (x1 , . . . , xi−1 , t , . . . , xd ) tends to ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) pointwise (as n → ∞) for almost all t ∈ I. Since every Cσ (n) is a shuffle of Min, one has
∂i Cσ (n) (x1 , . . . , xi−1 , t , . . . , xd ) ∈ {0, 1} almost everywhere, which implies that ∂i C (x1 , . . . , xi−1 , t , . . . , xd ) ∈ {0, 1} almost everywhere. Let x2 , . . . , xd ∈ I. In view of the Lemma 3.3 C ( x1 , . . . , xd ) =
x1
[0,x2 ]×···×[0,xd ]
0
dµx (x) dx.
Moreover, since C is absolutely continuous in each argument, it follows that C ( x1 , . . . , xd ) =
x1
∂1 C (x, x2 , . . . , xd )dx,
0
which implies that for almost all x ∈ I
µx ([0, x2 ] × · · · × [0, xd ]) =
[0,x2 ]×···×[0,xd ]
dµx (x)
= ∂1 C (x, x2 , . . . , xd ) ∈ {0, 1} . In view of Fubini Theorem it follows that in a subset of Id of measure 1 we have
µx ([0, x2 ] × · · · × [0, xd ]) ∈ {0, 1} . Therefore, µx is a Dirac measure for almost all x ∈ I and it concentrates the probability mass in a graph of a function of type (f2 , . . . , fd ). By repeating the above procedure along each of the axis, we may find that each fi is bijective. Thus, in view of Proposition 2.2(part b), C is a generalized shuffle of Min. Lemma 3.5. Let C ∈ S (Md ). Then there exists a a sequence (Cn )n of shuffles of Min that tends (in the M-convergence) to C , as n tends to ∞. Proof. Let C ∈ S (Md ). The Markov operator T associated with C is uniquely determined by the equality 1
Th(x)g (x)dx = 0
Id
g (x)h(x2 , . . . , xd )dµC (x, x2 , . . . , xd )
1766
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
for all h ∈ L∞ (Id−1 , µ′ ), µ′ (S ) = µC (I × S ) for any Borel set S ⊆ Id−1 , and g ∈ L∞ (I, λ). By applying Lemma 3.3 to the right hand side of the previous equality we get
Id
g (x)h(x2 , . . . , xd )dµC (x, x2 , . . . , xd ) =
1
g ( x)
Id−1
0
h(x2 , . . . , xd )dµx (x2 , . . . , xd ) dx.
It follows that for almost all x ∈ I Th(x) =
Id−1
h(x2 , . . . , xd )dµx (x2 , . . . , xd ).
Since C concentrates the mass on the set {(x, f2 (x), . . . , fd (x)): x ∈ I}, it follows that Th(x) = h(f2 (x), . . . , fd (x)) M
for almost all x ∈ I. Now, it is not difficult to see that Cn → C if, and only if, 1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx → 0, 0
where f2,n (x), . . . , fd,n are the measure-preserving transformations that support the graph where the mass of Cn is concentrated. In view of Lemma 3.2, for i ∈ {2, . . . , d} any fi can be approximated in the pointwise convergence by a sequence of measure-preserving transformations si,n that are piecewise linear. Since these functions are bounded, it can be derived from the Lebesgue Theorem that the convergence is guaranteed in the L1 -sense. Now, let h be a continuous function on I. Since h is also uniformly continuous on I, it follows that for ε > 0, there exists η > 0 such that, if maxi=2,...,n |xi − yi | < η, then |h(x2 , . . . , xd ) − h(y2 , . . . , yd )| < ε/2. Let η1 < min{η, ε/2, ε/(2d∥h∥)}. We may choose n ∈ N in such a way that 1
|fi (x) − si,n (x)|dx < η12 . 0
This implies that |fi (x) − si,n (x)| < η1 in a set Ai of measure 1 − η1 . Let A = ∩di=2 Ai and let B = I \ A. Then we have
1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx ≤ |h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx + ∥h∥dx < ε.
0
A
Since continuous functions are dense in the class of L
B
∞
functions it follows that
1
|h(f2 (x), . . . , fd (x)) − h(f2,n (x), . . . , fd,n (x))|dx → 0, 0
for all h ∈ L∞ (Id−1 , µ′ ), which is the desired assertion.
Proof of Proposition 3.1. (a) From Lemma 3.4 it follows that any sequence of shuffles of Min ∂ -converges to an element of S (Md ). Moreover, in view of Lemma 3.5 and the fact that M-convergence implies ∂ -convergence, any element of S (Md ) is a limit of shuffles of Min in the ∂ -convergence. This concludes the proof. (b) Analogously, from Lemma 3.4 and and the fact that M-convergence implies ∂ -convergence, it follows that any sequence of shuffles of Min M-converges to an element C that must be in S (Md ). Moreover, in view of Lemma 3.5, any element of S (Md ) is a a limit of shuffles of Min in the M-convergence. This concludes the proof. Remark 3.1. An example of a copula which is a generalized shuffle of Min, but not a shuffle of Min, can be given by the method described in Trutschnig (2012, Example 1). Specifically, we take a sequence of shuffles of Min by considering the probability mass of M2 in each of the squares described in Trutschnig (2012, Fig. 1). Then the limiting copula (in any of the above introduced convergences) is a generalized shuffle of Min. Acknowledgments We acknowledge the Reviewers for the careful reading and useful suggestions that have improved the manuscript. The first author acknowledges the support of the Free University of Bozen-Bolzano, School of Economics and Management, via the project ‘‘Risk and Dependence’’. The first and second author have been supported by the Ministerio de Ciencia e Innovación (Spain) under research project MTM2011-22394.
F. Durante, J.F. Sánchez / Statistics and Probability Letters 82 (2012) 1761–1767
1767
References Ambrosio, L., Gigli, N., Savaré, G., 2008. Gradient Flows in Metric Spaces and in the Space of Probability Measures, second ed. In: Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. Baker, R., 2008. An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99 (10), 2312–2327. Chou, S.-H., Nguyen, T.T., 1990. On Fréchet theorem in the set of measure preserving functions over the unit interval. Internat. J. Math. Math. Sci. 13 (2), 373–378. Darsow, W.F., Olsen, E.T., 2010. Characterization of idempotent 2-copulas. Note Mat. 30 (1), 193–235. de Amo, E., Díaz-Carrillo, M., Fernández-Sánchez, J., 2011. Measure-preserving functions and the independence copula. Mediterr. J. Math. 8 (3), 431–451. Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3–33. Durante, F., Fernández-Sánchez, J., 2010. Multivariate shuffles and approximation of copulas. Statist. Probab. Lett. 80 (23–24), 1827–1834. Durante, F., Sarkoci, P., Sempi, C., 2009. Shuffles of copulas. J. Math. Anal. Appl. 352 (2), 914–921. Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (Eds.), 2010. Copula Theory and its Applications. In: Lecture Notes in Statistics—Proceedings, vol. 198. Springer, Berlin, Heidelberg. Kolesárová, A., Mesiar, R., Mordelová, J., Sempi, C., 2006. Discrete copulas. IEEE Trans. Fuzzy Syst. 14 (5), 698–705. Kolesárová, A., Mesiar, R., Sempi, C., 2008. Measure-preserving transformations, copulæand compatibility. Mediterr. J. Math. 5 (3), 325–339. Lancaster, H.O., 1963. Correlation and complete dependence of random variables. Ann. Math. Statist. 34, 1315–1321. Mikusiński, P., Sherwood, H., Taylor, M.D., 1992. Shuffles of Min. Stochastica 13 (1), 61–74. Mikusiński, P., Taylor, M.D., 2009. Markov operators and n-copulas. Ann. Polon. Math. 96 (1), 75–95. Mikusiński, P., Taylor, M.D., 2010. Some approximations of n-copulas. Metrika 72 (3), 385–414. Nelsen, R.B., 2006. An Introduction to Copulas, second ed. In: Springer Series in Statistics, Springer, New York. Puccetti, G., Rüschendorf, L., 2012. Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236 (7), 1833–1840. Puccetti, G., Scarsini, M., 2010. Multivariate comonotonicity. J. Multivariate Anal. 101 (1), 291–304. Sancetta, A., Satchell, S., 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20 (3), 535–562. Siburg, K.F., Stoimenov, P.A., 2010. A measure of mutual complete dependence. Metrika 71 (2), 239–251. Sklar, A., 1959. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231. Trutschnig, W., 2011. On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384 (2), 690–705. Trutschnig, W., 2012. Some results on the convergence of (quasi-)copulas. Fuzzy Sets and Systems 191, 113–121. Vitale, R.A., 1990. On stochastic dependence and a class of degenerate distributions. In: Topics in Statistical Dependence (Somerset, PA, 1987). In: IMS Lecture Notes Monogr. Ser. Inst. Math. Statist., vol. 16, Hayward, CA, pp. 459–469. Vitale, R.A., 1991. Approximation by mutually completely dependent processes. J. Approx. Theory 66 (2), 225–228.