A topological proof of Sklar’s theorem

Applied Mathematics Letters (

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A topological proof of Sklar’s theorem Fabrizio Durante a,∗ , Juan Fernández-Sánchez b , Carlo Sempi c 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

c

Dipartimento di Matematica e Fisica ‘‘Ennio De Giorgi’’, Università del Salento, Lecce, Italy

article

info

Article history: Received 8 March 2013 Received in revised form 11 April 2013 Accepted 11 April 2013

abstract We present a proof of Sklar’s Theorem that uses topological arguments, namely compactness (under the weak topology) of the class of copulas and some density properties of the class of distribution functions. © 2013 Published by Elsevier Ltd

Keywords: Copula Sklar’s theorem

1. Introduction Sklar’s theorem is the fundamental step in the construction of multivariate stochastic models through a copula approach, i.e. by describing a joint probability distribution function (shortly, d.f.) in two steps: the knowledge of the univariate marginal d.f.’s and the copula, which captures the information about the dependence of the variables of interest. This approach has proved to be useful in a variety of fields of applied mathematics ranging from finance and insurance to geosciences: see, for instance, [1–6]. In view of its importance, Sklar’s theorem has been the object of several investigations reported in the literature with the aim of exploring its different facets. One possible strategy to prove this result consisted in constructing the so-called sub-copula associated with a joint d.f. and, then, extending it to a copula. This procedure was used for the first time in [7]. A similar construction may also be found in [8]. Another possible approach relies on probabilistic arguments based on suitable modifications of the probability integral transform; this is the approach used in [9] and, then, reconsidered in [10]. Others proofs based on specific tools have been obtained in [11–14]. Here we present a proof of Sklar’s Theorem that is based on topological arguments, basically the compactness (with respect to the weak topology) of the class of copulas and density properties of the class of d.f.’s. 2. Distribution functions and copulas In this section, we recall the preliminary notions that will be used in the sequel (see, for instance, [15]). Throughout this d

paper, d is a natural number with d ≥ 1, while Id := [0, 1]d and R := [−∞, +∞]d . d

For a function H : R → R, the H-volume VH of the box [a, b] := [a1 , b1 ] × [a2 , b2 ] × · · · × [ad , bd ] ,

∗

Corresponding author. Tel.: +39 0471013493; fax: +39 0471013009. E-mail addresses: [email protected] (F. Durante), [email protected] (J. Fernández-Sánchez), [email protected] (C. Sempi).

0893-9659/$ – see front matter © 2013 Published by Elsevier Ltd http://dx.doi.org/10.1016/j.aml.2013.04.005

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is defined by VH ([a, b]) :=



sign(v) H (v),

v

where the sum is taken over the 2d vertices v of the box [a, b]; here sign(v) = 1 if vj = aj for an even number of indices, while sign(v) = −1, otherwise. We consider the following definition. Definition 2.1. A d-dimensional distribution function (shortly, d.f.) is any function F : R properties:

d

→ I satisfying the following

d

(a) for every x ∈ R such that at least one argument of x is −∞, F (x) = 0, and F (+∞, +∞, . . . , +∞) = 1. (b) for every j ∈ {1, 2, . . . , d} and for all x1 , . . . , xj−1 , xj+1 , . . . , xd in R and t ∈ R, the function t → F (x1 , . . . , xj−1 , t , xj+1 , . . . , xd ) is right-continuous; (c) the F -volume VF of every box [a, b] is non-negative, i.e., VF ([a, b]) ≥ 0. For every d ≥ 1, let Dd be the set of all d-dimensional d.f.’s and let DdCont be its subset formed by the continuous d.f.’s. Notice that, since we consider distribution functions defined on the product of the extended real line, D1 is compact under the weak convergence (see for instance [16]). Definition 2.2. A d-copula is the restriction to Id of a distribution function whose univariate margins are uniformly distributed on I. The set of d-copulas is denoted by Cd . In the sequel, we shall use the fact that Cd is a compact space with respect to the weak topology (see, for instance, [14]). Our goal is to give another proof of the following result [17], which is the first, and harder part, of Sklar’s theorem. Theorem 2.1. Let H be a d-dimensional distribution function with univariate marginals F1 , F2 , . . . , Fd . Then there exists a copula d

C such that, for every x = (x1 , x2 , . . . , xd ) ∈ R , H (x1 , x2 , . . . , xd ) = C (F1 (x1 ), F2 (x2 ), . . . , Fd (xd )) .

(1)

3. Proof of Sklar’s theorem In proving Sklar’s theorem the main difficulty is encountered when considering d.f.’s that may present one or more discontinuities. In fact, when H is continuous, the proof can be easily obtained, as stated in the following result (see, for instance, [5]). Lemma 3.1. For every d-dimensional d.f. H with continuous marginals F1 , F2 , . . . , Fd there exists a unique copula C such that (1) holds. C is determined, for all u ∈ Id , via the formula



C (u) = H F1

[−1]

 (u1 ), F2[−1] (u2 ), . . . , Fd[−1] (ud ) ,

where, for i = 1, 2, . . . , d, Fi

[−1]

is the quasi-inverse of Fi (see [5]).

In order to overcome the difficulties that arise for non-continuous d.f.’s, we shall use some topological arguments. First, we need a preliminary lemma. Lemma 3.2. DdCont is dense in Dd with respect to the weak topology. Proof. For every x ∈ R, we set x,  x = 0, 1,



if 0 ≤ x ≤ 1, if x < 0, if 1 < x. d

For all H ∈ Dd and x ∈ R we consider the d.f. defined by

 H (x) := H (−cotan(π x1 ), . . . , −cotan(π xd )), where cotan(0) = +∞ and cotan(π ) = −∞. Moreover, we define H ∗ (x) =  H (x), if each xi ̸= 0, otherwise there exists i such that xi = 0 and we set H ∗ (x) := lim  H (x1 , . . . , xi−1 , t , xi+1 , . . . , xd ). t →0+

Analogously, one may define the values of H ∗ when more than one argument of x is equal to 0.

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The transformation ψ : Dd → Dd , ψ(H ) = H ∗ is a homeomorphism between Dd and ψ(Dd ) that maps a continuous d.f. into a continuous d.f.. In order to prove the assertion, it is sufficient to show that ψ(Dd )Cont , the set of continuous d.f.’s in ψ(Dd ), is dense in ψ(Dd ) (with respect to the weak topology). In fact, in such a case, it follows that, since ψ(Dd )Cont = ψ(DdCont ), DdCont is dense in Dd . To this end, let K be in ψ(Dd ). Consider intervals of I of the following type: Ii,n =

 i −1 n

, ni



for i = 1, . . . , n. It follows that Id can be decomposed into boxes of the form Bi1 ,...,id = Ii1 ,n × Ii2 ,n × · · · × Iid ,n .

(2)

Set VK (Bi1 ,...,id )

δi1 ,...,id =

λd (Bi1 ,...,id )

,

where λd is the Lebesgue measure. For every Borel subset A of Id , define

 

µn (A) :=

i1 ,...,id

A

δi1 ,...,id 1Bi1 ,...,id dλd .

Since µn (A) is the sum of integrals of positive functions on A, µn is a measure, which, by construction, is absolutely continuous with λd . It follows from the definition that µn (Bi1 ,...,id ) = VK (Bi1 ,...,id ) for every box of type (2). Let Kn be the d.f. associated with µn . Then Kn is continuous and verifies

 Kn

i1 n

,...,

id



n

 =K

i1 n

,...,

id



n

for every choice of i1 , . . . , id in {1, . . . , n}. Now, let x = (x1 , . . . , xd ) be a point in Id at which K is continuous. If x is such that each xi ∈



0,

ii 2m



for some natural

m,  then K2n (x) → K (x) for n → +∞. Otherwise, there exist some indices i1 , . . . , id ∈ {1, 2, . . . , 2 } such that each xj is in n

ij −1 2n

i

, 2jn , and  K

i1 − 1 2n

,...,

id − 1 2n



 = K2n

i1 − 1

id − 1



,..., ≤ K2n (x1 , . . . , xd ) 2n    id i1 id i1 , . . . , = K , . . . , . n n n n 2n

 ≤ K2n

2

2

2

2

Because K is continuous at x, it follows that K2n (x) → H (x) for n → +∞. It follows that K2n weakly converges to K and, hence, ψ(Dd )Cont is dense in ψ(Dd ).



Proof of Theorem 2.1. Define a mapping ϕ from Cd × (D1 )d to Dd such that, for every C ∈ Cd and for all F1 , . . . , Fd ∈ D1 , one has

ϕ (C , F1 , . . . , Fd ) = C (F1 , . . . , Fd ). In view of [18, Theorem 2], ϕ is continuous with respect to the product topology on Cd × (D1 )d and to the weak topology on Dd . d  Since Cd is compact and D1 is compact, Cd × (D1 ) is compact (with respect to the product topology). It follows that ϕ Cd × (D1 )d is also a compact set in Dd (in the weak topology) and, hence, is closed. Now, the class DdCont is dense in Dd with respect to the weak topology (see Lemma 3.2). Moreover, DdCont ⊆ ϕ(Cd ×(D1 )d ) (see Lemma 3.1). By density arguments, it follows that every F ∈ Dd belongs to ϕ(Cd × (D1 )d ), which is the desired assertion.  Acknowledgments The first author acknowledges the support of the Free University of Bozen-Bolzano, School of Economics and Management, via the project ‘‘Stochastic Models for Lifetimes’’. The first and second author have been supported by the Ministerio de Ciencia e Innovación (Spain) under research project MTM2011-22394.

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References [1] U. Cherubini, S. Mulinacci, F. Gobbi, S. Romagnoli, Dynamic Copula Methods in Finance, in: Wiley Finance Series, John Wiley & Sons Ltd., Chichester, 2012. [2] P. Jaworski, F. Durante, and W.K. Härdle (Eds.), Copulae in Mathematical and Quantitative Finance, in: Lecture Notes in Statistics—Proceedings, Springer, Berlin, Heidelberg, 2013. [3] P. Jaworski, F. Durante, W.K. Härdle, T. Rychlik (Eds.), Copula Theory and its Applications, in: Lecture Notes in Statistics—Proceedings, vol. 198, Springer, Berlin, Heidelberg, 2010. [4] J.-F. Mai, M. Scherer, Simulating Copulas, World Scientific, Singapore, 2012. [5] R.B. Nelsen, An Introduction to Copulas, second ed., in: Springer Series in Statistics, Springer, New York, 2006. [6] G. Salvadori, C. De Michele, N.T. Kottegoda, R. Rosso, Extremes in Nature. An Approach Using Copulas, in: Water Science and Technology Library, vol. 56, Springer, Dordrecht (NL), 2007. [7] B. Schweizer, A. Sklar, Operations on distribution functions not derivable from operations on random variables, Studia Math. 52 (1974) 43–52. [8] H. Carley, M.D. Taylor, A new proof of Sklar’s theorem, in: C.M. Cuadras, J. Fortiana, J.A. Rodriguez-Lallena (Eds.), Distributions with Given Marginals and Statistical Modelling, Kluwer Acad. Publ., Dordrecht, 2002, pp. 29–34. [9] D.S. Moore, M.C. Spruill, Unified large-sample theory of general chi-squared statistics for tests of fit, Ann. Statist. 3 (1975) 599–616. [10] L. Rüschendorf, On the distributional transform, Sklar’s theorem, and the empirical copula process, J. Statist. Plann. Inference 139 (11) (2009) 3921–3927. [11] A. Burchard, H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal. 233 (2) (2006) 561–582. [12] P. Deheuvels, A Bahadur–Kiefer representation for the empirical copula process, J. Math. Sci. (N. Y.) 163 (2009) 382–398. [13] E. de Amo, M. Díaz Carrillo, J. Fernández-Sánchez, Characterization of all copulas associated with non-continuous random variables, Fuzzy Sets and Systems 191 (2012) 103–112. [14] F. Durante, J. Fernández-Sánchez, C. Sempi, Sklar’s theorem obtained via regularization techniques, Nonlinear Anal. 75 (2) (2012) 769–774. [15] F. Durante, C. Sempi, Copula theory: an introduction, in: P. Jaworki, F. Durante, W. Härdle, T. Rychlik (Eds.), Copula Theory and its Applications, in: Lecture Notes in Statistics—Proceedings, vol. 198, Springer, Berlin, Heidelberg, 2010, pp. 3–31. [16] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Dover Publications, Mineola, NY, 2006. [17] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231. [18] C. Sempi, Convergence of copulas: critical remarks, Rad. Mat. 12 (2) (2004) 241–249.