A Topological Proof of Sklar's Theorem in Arbitrary Dimensions
aa r X i v : . [ m a t h . P R ] J a n A TOPOLOGICAL PROOF OF SKLAR’S THEOREM INARBITRARY DIMENSIONS
FRED ESPEN BENTH, GIULIA DI NUNNO, AND DENNIS SCHROERS
Abstract.
We prove Sklar’s theorem in infinite dimensions via a topological argu-ment and the notion of inverse systems. Introduction
Copulas are widely used and well known concepts in the realm of statistics and proba-bility theory. The keystone of the theory is Sklar’s theorem and there is a vast literaturesolely focussing on different proofs of this fundamental result. Among others there areproofs based on the distributional transform in [10] and [4] and earlier already in [8],based on mollifiers in [5] or the constructive approach by the extension of subcopulas,as it was proved for the bivariate case in [11] and for the general multivariate case in[14] or [3].The naive transfer of the subcopula-approach to an infinite-dimensional setting ap-pears to be challenging, since, after the extension of the subcopulas corresponding tothe finite-dimensional laws of an infinite-dimensional distribution, one would also haveto check that this construction meets the necessary consistency conditions. In contrast,and besides the approach via distributional transforms (as extended to an infinite di-mensional setting in [2]), a nonconstructive proof based on topological arguments in [6]is naturally in tune with an infinite dimensional setting.In this paper, we will therefore adopt this ansatz and prove Sklar’s theorem in infinitedimensions by equipping the space of copulas with an inverse-limit topology that makesit compact and the operation between marginals and copulas induced by Sklar’s theoremcontinuous. The compactness of copulas is described as "folklore" in [5] for the finitedimensional case, which is why the transfer to arbitrary dimensions is desirable.2.
Short Primer on Topological Inverse Systems
We will frequently use the notation ¯ R for the extended real line [ −∞ , ∞ ] . For anymeasure µ on a measurable space ( B, B ) and a measurable function f : ( B, B ) → ( A, A ) into another measurable space ( A, A ) we denote by f ∗ µ the pushforward measure withrespect to f given by f ∗ µ ( S ) := µ ( f − ( S )) for all S ∈ A . For I an arbitrary indexset, B = ¯ R I and B = ⊗ i ∈ I B ( ¯ R ) , we use the shorter notations π J ∗ µ =: µ J for a subset J ⊆ I and π { i }∗ µ =: µ i for an element i ∈ I , where π J denotes the canonical projectionon R J . If J ⊂ I is finite, we denote the corresponding finite dimensional cumulativedistribution functions by F µ J or F µ i respectively, where in the latter we used J = { i } .We use the notation I for the set consisting of all finite subsets of I . Moreover, for Date : January 22, 2021.2010
Mathematics Subject Classification.
Primary 60E05, 62H05; Secondary 62H20, 28C20.
Key words and phrases.
Copulas, Sklar’s theorem, Topological inverse limits, Infinite Dimensions,Compactness. a one-dimensional Borel measure µ i on R , we use the notation F [ − µ i for the quantilefunctions F [ − µ i ( u ) := inf { x ∈ ( −∞ , ∞ ) : F µ i ( x ) ≥ u } . (2.1)We will refer to the one dimensional distributions µ i , i ∈ I and equivalently F µ i , i ∈ I as marginals of the measure µ . We denote the set of all probability measures on ( ¯ R I , ⊗ i ∈ I B ( ¯ R )) by P ( ¯ R I ) . Moreover, for two topological spaces X, Y we write X ∼ = Y if they are homeomorphic.The remainder of the section is mainly based on [9]. Let X J be a set for each J ∈ I and ( P J ,J : X J → X J ) for J ⊆ J , with J , J ∈ I a family of mappings, also called projections, such that(i) P J,J = id J is the identity mapping for all J ∈ I , and(ii) P J ,J = P J ,J ◦ P J ,J for all J ⊆ J ⊆ J in I .The system ( X J , P J ,J , I ) := (cid:18) ( X J ) J ∈I , (( P J ,J : X J → X J ) J ⊆ J J ,J ∈I ) (cid:19) is called an inverse system (over the partially ordered set I ). If ( X J , τ J ) are topologicalspaces for each J ∈ I and ( P J ,J ) are continuous for all J ⊆ J with J , J ∈ I , wecall ( X J , τ J , P J ,J , J ∈ I ) := (cid:18) ( X J , τ J ) J ∈I , (( P J ,J : X J → X J ) J ⊆ J J ,J ∈I ) (cid:19) a topological inverse system . A topological inverse limit of this inverse system is a space X together with continuous mappings P J : X X J , J ∈ I , such that P J ,J P J = P J for all J ⊆ J in I (that is, the mappings are compatible ) and the following universalproperty holds: Whenever there is a topological space Y , such that there are continuousmappings ( ψ J : Y → X J ) J ∈I which are compatible, i.e., P J ,J ψ J = ψ J for all J ⊆ J in I , then there exists a unique continuous mapping Ψ : Y → X, (2.2)with the property P J Ψ = ψ J for all J ∈ I . We have that ( x = ( x J ) J ∈I ∈ Y J ∈I X J : P J ,J ( π J ( x )) = π J ( x ) for J ⊆ J ) ⊆ Y J ∈I X J (2.3)equipped with the subspace topology with respect to the product topology is an in-verse limit of the topological inverse system, induced by the canonical projections π J ′ (( x J ) J ∈I ) = x J ′ . Each topological inverse limit is homeomorphic to this space andtherefore to every topological inverse limit (See the proof of Theorem 1.1.1 in [9]). Wewrite lim ← X J ⊆ Q J ∈I X J for the inverse limit as a subset of the product space andwe equip it throughout with the induced subspace topology. Lemma 2.1.
Let ( X J , τ J , π J ,J ) be a topological inverse system (over the poset I ) ofHausdorff spaces. Then lim ← X J is a closed subset of Q J ∈I X J with respect to theproduct topology.Proof. See [9, Lemma 1.1.2]. (cid:3)
TOPOLOGICAL PROOF OF SKLAR’S THEOREM IN ARBITRARY DIMENSIONS 3
Lemma 2.2.
Let X be a compact Hausdorff space and ( X J , τ J , π J ,J ) be a topologicalinverse system of compact Hausdorff spaces. Let ψ J : X → X J , J ∈ I be a family ofcompatible surjections and Ψ the induced mapping. Then either lim ← X J = ∅ or Ψ( X ) is dense in lim ← X J .Proof. See [9, Corollary 1.1.7]. (cid:3) Copulas and Sklar’s Theorem
As they are cumulative distribution functions, copulas in finite dimension have a one-to-one correspondence to probability measures. In infinite dimensions we will thereforework with the notion of copula measures as introduced in [2].
Definition 3.1.
A copula measure (or simply copula) on ¯ R I is a probability measure C ∈ P ( ¯ R I ) , such that its marginals C i are uniformly distributed on [0 , . We will denotethe space of copula measures on ¯ R I by C ( ¯ R I ) . Sklar’s theorem as stated below was proved in [2] by following the arguments for thefinite dimensional assertion in [10]. Here we give an alternative proof for the infinitedimensional setting using a topological argument as in [6].
Theorem 3.2 (Sklar’s Theorem) . Let µ ∈ P ( ¯ R I ) be a probability measure with marginalone-dimensional distributions µ i , i ∈ I . There exists a copula measure C , such that foreach J ∈ I , we have F C J (cid:16)(cid:0) F µ j ( x j ) (cid:1) j ∈ J (cid:17) = F µ J (( x j ) j ∈ J ) (3.1) for all ( x j ) j ∈ J ∈ ¯ R J . Moreover, C is unique if F µ i is continuous for each i ∈ I .Vice versa, let C be a copula measure on ¯ R I and let ( µ i ) i ∈ I be a collection of (one-dimensional) Borel probability measures over ¯ R . Then there exists a unique probabilitymeasure µ ∈ P ( ¯ R I ) , such that (3.1) holds. Topological Properties of Copulas and a Proof of Sklar’s Theorem
The collection ( P ( ¯ R J ) , J ∈ I ) , where each P ( ¯ R J ) is considered as a topological spacewith the topology of weak convergence, is a topological inverse system with the pro-jections π J ,J ( µ J ) = ( µ J ) J for µ J ∈ P ( ¯ R J ) and J , J ∈ I , J ⊆ J . Moreover,observe that each P ( ¯ R J ) is a Hausdorff space, since it is metrizable by the Prohorovmetric (c.f. [12, Theorem 4.2.5]). The space lim ← P ( ¯ R J ) ⊂ Q J ∈I P ( ¯ R J ) of consistentfamilies of probability measures is a topological inverse limit, equipped with the corre-sponding inverse limit topology. The space of probability measures on ⊗ i ∈ I B ( R ) has viaits finite-dimensional distributions a one-to-one correspondence with this family of con-sistent finite-dimensional distributions, and hence there is a natural bijection between lim ← P ( ¯ R J ) and P ( ¯ R I ) .We equip the space P ( ¯ R I ) with the topology of weak convergence of the finite dimen-sional distributions , which we define as follows: Definition 4.1.
The topology of convergence of the finite dimensional distributions on P ( ¯ R I ) is defined as the topology such that P ( ¯ R I ) ∼ = lim ← P ( ¯ R J ) . P ( ¯ R I ) with this topology is by definition a topological inverse limit. Define also lim ← C ( ¯ R J ) := lim ← P ( ¯ R J ) ∩ Q J ∈I C ( ¯ R J ) . Certainly, we have C (cid:0) ¯ R I (cid:1) ∼ = lim ← C (cid:0) ¯ R J (cid:1) (4.1) FRED ESPEN BENTH, GIULIA DI NUNNO, AND DENNIS SCHROERS with the corresponding topologies.The following result contains among other things the topological proof of Sklar’stheorem 3.2.
Theorem 4.2.
The following statements hold.(1) P ( ¯ R I ) with the topology of weak convergence of the finite dimensional distribu-tions is a Hausdorff space(2) The space of consistent copulas C ( ¯ R I ) is compact with respect to the topology ofconvergence of finite dimensional distributions.(3) For a copula measure C on ¯ R I and (one-dimensional) Borel probability measures ( µ i ) i ∈ I over ¯ R the push-forward measure µ := (( F [ − µ i ) i ∈ I ) ∗ C (4.2) satisfies (3.1) .(4) If we equip C ( ¯ R I ) × Q i ∈ I P ( ¯ R ) with the product topology of weak convergence oneach P ( ¯ R ) and the topology of convergence of the finite dimensional distributionson C ( ¯ R I ) and P ( ¯ R I ) , then the mapping Φ : C ( ¯ R I ) × Q i ∈ I P ( ¯ R ) → P ( ¯ R I ) givenby Φ( C, ( µ i ) i ∈ I ) := (( F [ − µ i ) i ∈ I ) ∗ C is continuous and surjective. In particular, Sklar’s theorem holds.Proof. (1) Since products of Hausdorff spaces are Hausdorff and P ( ¯ R I ) is homeomorphicto a subset of a product of Hausdorff spaces, it is Hausdorff.(2) We know by [5, Thm. 3.3] that every C ( ¯ R J ) is compact with respect to thetopology of weak convergence on P ( ¯ R J ) . Tychonoff’s Theorem guarantees also that Q J ∈I C ( ¯ R J ) is compact with respect to the product topology on Q J ∈I P ( ¯ R J ) . There-fore, as lim ← P ( ¯ R J ) is closed by Lemma 2.1, we obtain that C ( ¯ R I ) is compact, since it ishomeomorphic to an intersection of a closed and a compact set in the product topology.(3) This corresponds to the second part of Sklar’s theorem and the proof can beconducted analogously to the one in [2]. Therefore, it is enough to see that (cid:0)(cid:2) , F µ j ( x j ) (cid:3)(cid:1) j ∈ J \ (cid:18)(cid:16) F [ − µ j (cid:17) − ( −∞ , x ] (cid:19) j ∈ J is a C J -nullset for all ( x j ) j ∈ J ∈ ¯ R J , J ∈ I , since then we immediately obtain C J (cid:18)(cid:16) F [ − µ j (cid:17) − ( −∞ , x ] (cid:19) j ∈ J ! = C J (cid:16)(cid:0) [0 , F µ j ( x j )] (cid:1) j ∈ J (cid:17) = F C J (cid:16) F µ j (( x j )) j ∈ J (cid:17) . (4) Define φ J : C ( ¯ R I ) × Q i ∈ I P ( ¯ R ) → P ( ¯ R J ) by φ J ( C, ( µ i ) i ∈ I ) := Φ( C, ( µ i ) i ∈ I ) J , which is well defined by (3). Since the finite-dimensional distributions of a law areconsistent, ( φ J , J ∈ I ) forms a compatible family. Define analogously for J ∈ I also ˜ φ J : C ( ¯ R J ) × Q j ∈ J P ( ¯ R ) → P ( ¯ R J ) by ˜ φ J ( C J , ( µ j ) j ∈ J ) = ( F [ − µ j ) j ∈ J ) ∗ C J . This is by Sklar’s theorem in finite-dimensions surjective and by [13, Thm. 2] alsocontinuous. Hence φ J = ˜ φ J π J is continuous and surjective, since both, ˜ φ J and π J are. Φ must be the uniquely induced continuous mapping by the family ( φ J , J ∈ I ) by theuniversality property of the inverse limit. Moreover, since by [12, Corollary 4.2.6] P ( ¯ R ) is TOPOLOGICAL PROOF OF SKLAR’S THEOREM IN ARBITRARY DIMENSIONS 5 compact and by (2) also C ( ¯ R I ) is compact, we have that C ( ¯ R I ) × Q i ∈ I P ( ¯ R ) is compact byTychonoff’s theorem. The continuity of Φ implies therefore that Φ( C ( ¯ R I ) × Q i ∈ I P ( ¯ R )) iscompact, hence closed. Since moreover Lemma 2.2 implies that Φ( C ( ¯ R I ) × Q i ∈ I P ( ¯ R )) is dense, we obtain that Φ is surjective and therefore also the first part of Sklar’stheorem holds. The uniqueness of the copulas in the case of continuous marginalsfollows immediately by Sklar’s theorem in finite dimensions via the uniqueness of thefinite dimensional distribution of the corresponding copula measure. (cid:3) Observe that since P ( ¯ R J ) is a locally convex Hausdorff space with respect to thetopology of weak convergence for each J ∈ I , we obtain that also the inverse limit P ( ¯ R I ) is locally convex, as it is isomorphic to a subset of the product Q J ∈I P ( R J ) of locally convex Hausdorff spaces. Hence, as mentioned for instance in [7, p.30], since C ( R I ) is convex, we have that it is the closure of its extremal points by the Krein-Milmantheorem. As mentioned in [1] this implies that sup C ∈C ( R I ) g ( C ) = sup C ∈ ext ( C ( R I )) g ( C ) where ext ( C ( R I )) denotes the set of extremal points of C ( R I ) and g : C ( R I ) R is aconvex function. Acknowledgements.
This research was funded within the project STORM: Stochas-tics for Time-Space Risk Models, from the Research Council of Norway (RCN). Projectnumber: 274410.
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FRED ESPEN BENTH, GIULIA DI NUNNO, AND DENNIS SCHROERS (Fred Espen Benth)
Department of Mathematics, University of Oslo, Norway
Email address , Fred Espen Benth: [email protected] (Giulia Di Nunno)
Department of Mathematics, University of Oslo, Norway
Email address , Giulia Di Nunno: [email protected] (Dennis Schroers)
Department of Mathematics, University of Oslo, Norway
Email address , Dennis Schroers:, Dennis Schroers: