How to prove Sklar's Theorem

How to prove Sklar’s Theorem Fabrizio Durante, Juan Fern´andez-S´anchez and Carlo Sempi

Abstract In this contribution we stress the importance of Sklar’s theorem and present a proof of this result that is based on the compactness of the class of copulas (proved via elementary arguments) and the use of mollifiers. More details about the procedure can be read in a recent paper by the authors.

1 Introduction The concept of copula was introduced by A. Sklar [20] in order to describe in a convenient way the class of distribution functions with given marginals. After Sklar’s paper, copulas have been used in the study of the aggregation of information at different levels and under diverse aspects. Specifically, just to refer to a few examples, the following directions have been pursued in the literature. • Firstly, copulas are useful in order to aggregate (e.g., join) univariate marginals distribution functions into multivariate models of distribution functions that have more flexibility (tail dependency, asymmetry) than standard (e.g., Gaussian) models (see, for instance, [10, 11, 12]). • The copula structure of a random vector is a key ingredient in order to estimate and make inference about the risk connected with the aggregation of different Fabrizio Durante School of Economics and Management, Free University of Bozen–Bolzano, Italy e-mail: [email protected] Juan Fern´andez-S´anchez Grupo de Investigaci´on de An´alisis Matem´atico, Universidad de Almer´ıa, La Ca˜nada de San Urbano, Almer´ıa, Spain e-mail: [email protected] Carlo Sempi Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Universit`a del Salento, Lecce, Italy e-mail: [email protected]

1

2

Fabrizio Durante, Juan Fern´andez-S´anchez and Carlo Sempi

random variables, especially, when they represent losses and/or assets of financial markets (see, for instance, [3, 7, 18]). In particular, upper and lower bounds for risk measures of random vectors can be derived by using Hoeffding-Fr´echet bounds for copulas or similar inequalities (see, for instance, [21]). • Aggregation of different inputs into a single numerical output in order to handle decisions, possibly in presence of general uncertainty models (see, for instance, [8, 9]) has benefited from the use of copulas, which has allowed robust (i.e., Lipschitzian) aggregation procedures. Moreover, recent trends in non–additive integrals have used copulas in order to aggregate preferences in a convenient way [13, 14]. These new directions in copula theory underline the need for underpinning the basic ideas about copulas and developing theoretical research about the foundations of the concept and its implications in different areas of mathematics. Here we aim at discussing the first issue about copula theory, namely the representation of random vectors in terms of a copula and the univariate margins. Such result goes under the name of “Sklar’s Theorem” since it was discovered by Abe Sklar [20] in its seminal paper of 1959. In particular, we present here a new proof of Sklar’s Theorem that underlines some non–standard perspectives on the problem (for more details, see [6]).

2 Sklar’s Theorem Before proceeding we briefly recall the definition of a copula; a d–copula C is the restriction to the unit hypercube [0, 1]d of a distribution function (d.f., for short) d on R that has uniform univariate margins on [0, 1]d . Therefore (a) C(u) = 0 whenever at least one of the components of u = (u1 , . . . , ud ) equals zero, (b) C(1, . . . , 1,t, 1, . . . , 1) = t, when all the components of u equal 1, with the possible exception of the j–th one and (c) the C volume VC of every d–box [a, b] := [a1 , b1 ] × · · · × [ad , bd ] contained in [0, 1]d is non-negative, namely VC ([a, b]) := ∑ sign(v)C(v) ≥ 0 , v

where the sum is taken over the 2d vertices v of the rectangle [a, b] and ( 1, if v j = a j for an even number of indices, sign(v) = −1 , if v j = a j for an odd number of indices. As a consequence of the definition of copula, one has that every d–copula C satisfies the Lipschitz condition |C(x) −C(y)| ≤ kx − yk1 ,

How to prove Sklar’s Theorem

3

where k · k1 is the `1 –norm on [0, 1]d . For more details, see the monographs [11, 12, 16]. Nowadays, Sklar’s theorem represents the building block of the modern theory of multivariate d.f.’s. It can be formulated as follows. Theorem 1 (Sklar’s Theorem). Let (X1 , . . . , Xd ) be a random vector with joint d.f. H and univariate marginals F1 , . . . , Fd . Then there exists a copula C : [0, 1]d → [0, 1] such that, for all x = (x1 , . . . , xd ) ∈ Rd , H(x1 , . . . , xd ) = C (F1 (x1 ), . . . , Fd (xd )) . C is uniquely determined on Range(F1 ) × · · · × Range(Fd ) and, hence, it is unique when F1 , . . . , Fd are continuous. Because of its importance in applied probability and statistics, Sklar’s theorem has received a great deal of attention and has been proved several times (and with different techniques). It was announced, but not proved, by [20], who provided (together with Schweizer) a complete proof in [19]. Other proofs of Sklar’s Theorem have been given in the literature. These are based either on analytical arguments, trying to extend a so-called sub-copula to a copula in some ways (see [2, 4]), or on probabilistic techniques, based on the modifications of the probability integral transform (see [5, 15, 17]). Remark 1. It should be stressed that most of the proofs of Sklar’s Theorem have slightly different settings. In the original proof by Schweizer and Sklar [19], the d authors considered d.f.’s defined on R . In most of the probabilistic approaches to Sklar’s Theorem [15, 17] (included the above described method), d.f.’s are defined on Rd (with some suitable margins). Depending on which settings is involved, both definitions have their pros and cons. Actually, in the case of continuous d.f.’s, Sklar’s Theorem admits an easy proof, since the following result holds. Lemma 1. For every d–dimensional d.f. H with continuous marginals F1 , . . . , Fd there exists a unique d–copula C such that, for all x = (x1 , . . . , xd ) ∈ Rd , H(x) = C (F1 (x1 ), . . . , Fd (xd )) .

(1)

Such a C is determined, for all u ∈ ]0, 1[d , via the formula   [−1] [−1] C(u) = H F1 (u1 ), . . . , Fd (ud ) , [−1]

where, for i ∈ {1, . . . , d} Fi Fi (x) ≥ t}.

[−1]

is the quasi–inverse of Fi defined by Fi

(t) := inf{x :

The hard part of Sklar’s Theorem consists in the extension to the case when at least one of the marginals has a discrete component.

4

Fabrizio Durante, Juan Fern´andez-S´anchez and Carlo Sempi

Here we aim at presenting a different proof of Sklar’s Theorem following [6]. This proof is based on the compactness of the class of copulas that is usually proved by Ascoli–Arzel`a Theorem, which we circumvent by the following result. Theorem 2. The set of all copulas Cd is a compact subset in the class of all continuous functions from [0, 1]d to [0, 1]. Proof. Since [0, 1]d is a metric space under the metric of the norm of uniform convergence, it is enough to show that Cd is sequentially compact. (1) (d) Let (Cn )n be a sequence of copulas in Cd and let (x j ) = (x j , . . . , x j ) be a dense sequence of points in [0, 1]2 ; for instance take the sequence   id i1 ,..., (n ∈ N; i1 , . . . , id = 0, 1, . . . , n) , n n and order it according to lexicographical order. Consider the bounded sequence of real numbers (Cn (x1 )); then there exists a convergent subsequence C1,n (x1 ). For the same reason, the sequence (C1,n (x2 )) contains a convergent subsequence (C2,n (x2 )). Notice that both sequences (C2,n (x1 )) and (C2,n (x2 )) converge. Proceeding in this way one constructs, for every k ≥ 1, a subsequence (Ck,n ) of (Ck−1,n ) that converges at the points x j with j ≤ k. Consider now the diagonal sequence (Ck,k )k∈N ; this converges at every point of the sequence (x j ). Define the function C : (x j ) j∈N → [0, 1] via C(x j ) := lim Ck,k (x j ) . k→+∞

Since (Ck,k )k∈N satisfies the Lipschitz condition |Ck,k (xi ) − Ck,k (x j )| ≤ kxi − x j k1 , one has, on taking the limit as k goes to +∞, C(xi ) −C(x j ) ≤ kxi − x j k1 , which proves that C is uniformly continuous on the sequence (xn ); and since this latter is dense in [0, 1]d , the definition of C can be extended by continuity to the whole of [0, 1]d . It immediately follows from its very definition that C satisfies the boundary conditions of a copula. Consider now the d–box [a, b], where the points a and b belong to the sequence (xn ). Then the C–Volume of [a, b] is given by VC ([a, b]) := ∑ sign(v)C(v) v

= lim

lim VCk,k ([a, b]) ≥ 0 . ∑ sign(v)Ck,k (v) = k→+∞

k→+∞ v

(s)

Finally, consider a d–box [a, b] contained in [0, 1]d and 2d subsequences (vn ) of (x j ) (s = 1, 2, . . . , 2d ) that converge to the vertices v(s) of [a, b]. The continuity of C yields

How to prove Sklar’s Theorem

5

2d

VC ([a, b]) =

∑ sign(v(s) )C(v(s) ) = lim

n→+∞

j=1

2d

(s)

∑ sign(vn

(s)

)C(vn ) ≥ 0 ,

j=1

which proves that C is indeed a d–copula. t u

3 Proof of Sklar’s Theorem by means of mollifiers Here we sketch the main arguments presented in [6]. Let Br (a) denote the open ball in Rd of centre a and radius r and consider the function ϕ : Rd → R defined by   1 ϕ(x) := k exp 1B1 (0) (x), |x|2 − 1 where the constant k is such that the L1 norm kϕk1 of ϕ is equal to 1. Further, for ε > 0, define ϕε : Rd → R by ϕε (x) :=

1 x ϕ . ε εd

It is known (see, e.g., [1, Chapter 4]) that ϕε belongs to C∞ (Rd ), and that its support is the closed ball Bε (0). Functions like ϕε are sometimes called mollifiers. If a d–dimensional d.f. H is given, then the convolution Z

Hn (x) :=

H(x − y) ϕ1/n (y) dy =

Rd

Z

ϕ1/n (x − y) H(y) dy

(2)

Rd

is well defined for every x ∈ Rd and finite; in fact, Hn is bounded below by 0 and above by 1. The following facts hold. Lemma 2. For every d–dimensional d.f. H and for every n ∈ N, the function Hn defined by (2) is a d–dimensional continuous d.f.. Lemma 3. If H is continuous at x ∈ Rd , then limn→+∞ Hn (x) = H(x). We are now ready for the final step. Proof of Sklar’s Theorem. We only sketch the main idea, the details being presented in [6]. For any given d.f. H, construct, for every n ∈ N, the continuous d.f. Hn defined by eq. (2); its marginals Fn,1 , . . . , Fn,d are also continuous; therefore, Lemma 1 ensures that there exists a d–copula Cn such that
Hn (x) = Cn Fn,1 (x1 ), . . . , Fn,d (xd ) .

6

Fabrizio Durante, Juan Fern´andez-S´anchez and Carlo Sempi

Because of the compactness of Cd , there exists a subsequence (Cn(k) )k∈N ⊂ (Cn )n∈N that converges to a copula C. Now, the thesis follows by showing that such a C is actually a possible copula of H.  Acknowledgements The first author acknowledges the support of Free University of BozenBolzano, 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 Innovaci´on (Spain) under research project MTM2011-22394.

References 1. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011) 2. Carley, H., Taylor, M.D.: 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, pp. 29–34. Kluwer Acad. Publ., Dordrecht (2002) 3. Cherubini, U., Mulinacci, S., Gobbi, F., Romagnoli, S.: Dynamic Copula methods in finance. Wiley Finance Series. John Wiley & Sons Ltd., Chichester (2012) 4. de Amo, E., D´ıaz-Carrillo, M., Fern´andez-S´anchez, J.: Characterization of all copulas associated with non-continuous random variables. Fuzzy Sets and Systems 191, 103–112 (2012) 5. Deheuvels, P.: Caract´erisation compl`ete des lois extrˆemes multivari´ees et de la convergence des types extrˆemes. Publ. Inst. Stat. Univ. Paris 23(3-4), 1–36 (1978) 6. Durante, F., Fern´andez-S´anchez, J., Sempi, C.: Sklar’s theorem obtained via regularization techniques. Nonlinear Anal. 75(2), 769–774 (2012) 7. Embrechts, P., Puccetti, G.: Risk aggregation. In: P. Jaworski, F. Durante, W. H¨ardle, T. Rychlik (eds.) Copula Theory and its Applications, Lecture Notes in Statistics - Proceedings, vol. 198, pp. 111–126. Springer, Berlin Heidelberg (2010) 8. Figueira, J., Greco, S., Ehrgott, M.: Multiple Criteria Decision Analysis: State of the Art Surveys. Springer Verlag, Boston, Dordrecht, London (2005) 9. Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions. Encyclopedia of Mathematics and its Applications (No. 127). Cambridge University Press, New York (2009) 10. Jaworski, P., Durante, F., H¨ardle, W. (eds.): Copulae in Mathematical and Quantitative Finance. Lecture Notes in Statistics - Proceedings. Springer, Berlin Heidelberg (2013) 11. Jaworski, P., Durante, F., H¨ardle, W., Rychlik, T. (eds.): Copula Theory and its Applications, Lecture Notes in Statistics - Proceedings, vol. 198. Springer, Berlin Heidelberg (2010) 12. Joe, H.: Multivariate models and dependence concepts, Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997) 13. Klement, E.P., Koles´arov´a, A., Mesiar, R., Stupnanov´a, A.: A generalization of universal integrals by means of level dependent capacities. Knowledge-Based Systems 38, 14–18 (2013) 14. Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Systems 18(1), 178–187 (2010) 15. Moore, D.S., Spruill, M.C.: Unified large-sample theory of general chi-squared statistics for tests of fit. Ann. Statist. 3, 599–616 (1975) 16. Nelsen, R.B.: An introduction to copulas, second edn. Springer Series in Statistics. Springer, New York (2006) 17. R¨uschendorf, L.: On the distributional transform, Sklar’s Theorem, and the empirical copula process. J. Statist. Plan. Infer. 139(11), 3921–3927 (2009) 18. Salvadori, G., De Michele, C., Durante, F.: On the return period and design in a multivariate framework. Hydrol. Earth Syst. Sci. 15, 3293–3305 (2011) 19. Schweizer, B., Sklar, A.: Operations on distribution functions not derivable from operations on random variables. Studia Math. 52, 43–52 (1974)

How to prove Sklar’s Theorem

7

20. Sklar, A.: Fonctions de r´epartition a` n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959) 21. Tankov, P.: Improved fr´echet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 48(2), 389–403 (2011)