Copulas are widely used and well known concepts in the realm of statistics and probability theory. The keystone of the theory is Sklar’s theorem and there is a vast literature solely focussing on different proofs of this fundamental result. Among others there are proofs based on the distributional transform in[RÜS09] and [DEH09] and earlier already in [MS75], based on mollifiers in [DFS12] or the constructive approach by the extension of subcopulas, as it was proved for the bivariate case in [SS74] and for the general multivariate case in [SKL96] or [CT02].
The naive transfer of the subcopula-approach to an infinite-dimensional setting appears to be challenging, since, after the extension of the subcopulas corresponding to the finite-dimensional laws of an infinite-dimensional distribution, one would also have to check that this construction meets the necessary consistency conditions. In contrast, and besides the approach via distributional transforms (as extended to an infinite dimensional setting in [BDS20]), a nonconstructive proof based on topological arguments in [DFS13] is naturally in tune with an infinite dimensional setting.
In this paper, we will therefore adopt this ansatz and prove Sklar’s theorem in infinite dimensions by equipping the space of copulas with an inverse-limit topology that makes it compact and the operation between marginals and copulas induced by Sklar’s theorem continuous. The compactness of copulas is described as "folklore" in [DFS12] for the finite dimensional case, which is why the transfer to arbitrary dimensions is desirable.
2. Short Primer on Topological Inverse Systems
We will frequently use the notation for the extended real line . For any measure on a measurable space and a measurable function into another measurable space we denote by the pushforward measure with respect to given by for all . For an arbitrary index set, and , we use the shorter notations for a subset and for an element , where denotes the canonical projection on . If
is finite, we denote the corresponding finite dimensional cumulative distribution functions byor respectively, where in the latter we used . We use the notation for the set consisting of all finite subsets of . Moreover, for a one-dimensional Borel measure on , we use the notation
for the quantile functions
We will refer to the one dimensional distributions and equivalently as marginals of the measure
. We denote the set of all probability measures onby . Moreover, for two topological spaces we write if they are homeomorphic.
The remainder of the section is mainly based on [RZ10]. Let be a set for each and
a family of mappings, also called projections, such that
is the identity mapping for all , and
for all in .
is called an inverse system (over the partially ordered set ). If are topological spaces for each and are continuous for all with , we call
a topological inverse system. A topological inverse limit of this inverse system is a space together with continuous mappings , such that for all in (that is, the mappings are compatible) and the following universal property holds: Whenever there is a topological space , such that there are continuous mappings which are compatible, i.e., for all in , then there exists a unique continuous mapping
with the property for all . We have that
equipped with the subspace topology with respect to the product topology is an inverse limit of the topological inverse system, induced by the canonical projections . Each topological inverse limit is homeomorphic to this space and therefore to every topological inverse limit (See the proof of Theorem 1.1.1 in [RZ10]). We write for the inverse limit as a subset of the product space and we equip it throughout with the induced subspace topology.
Let be a topological inverse system (over the poset ) of Hausdorff spaces. Then is a closed subset of with respect to the product topology.
See [RZ10, Lemma 1.1.2]. ∎
Let be a compact Hausdorff space and be a topological inverse system of compact Hausdorff spaces. Let be a family of compatible surjections and the induced mapping. Then either or is dense in .
See [RZ10, Corollary 1.1.7]. ∎
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 therefore work with the notion of copula measures as introduced in [BDS20].
A copula measure (or simply copula) on is a probability measure , such that its marginals are uniformly distributed on
are uniformly distributed on. We will denote the space of copula measures on by .
Sklar’s theorem as stated below was proved in [BDS20] by following the arguments for the finite dimensional assertion in [RÜS09]. Here we give an alternative proof for the infinite dimensional setting using a topological argument as in [DFS13].
Theorem 3.2 (Sklar’s Theorem).
Let be a probability measure with marginal one-dimensional distributions . There exists a copula measure , such that for each , we have
for all . Moreover, is unique if is continuous for each . Vice versa, let be a copula measure on and let be a collection of (one-dimensional) Borel probability measures over . Then there exists a unique probability measure , such that (3.1) holds.
4. Topological Properties of Copulas and a Proof of Sklar’s Theorem
The collection , where each is considered as a topological space with the topology of weak convergence, is a topological inverse system with the projections for and , . Moreover, observe that each is a Hausdorff space, since it is metrizable by the Prohorov metric (c.f. [SS05, Theorem 4.2.5]). The space of consistent families of probability measures is a topological inverse limit, equipped with the corresponding inverse limit topology. The space of probability measures on has via its finite-dimensional distributions a one-to-one correspondence with this family of consistent finite-dimensional distributions, and hence there is a natural bijection between and .
We equip the space with the topology of weak convergence of the finite dimensional distributions, which we define as follows:
The topology of convergence of the finite dimensional distributions on is defined as the topology such that .
with this topology is by definition a topological inverse limit. Define also . Certainly, we have
with the corresponding topologies.
The following result contains among other things the topological proof of Sklar’s theorem 3.2.
The following statements hold.
with the topology of weak convergence of the finite dimensional distributions is a Hausdorff space
The space of consistent copulas is compact with respect to the topology of convergence of finite dimensional distributions.
For a copula measure on and (one-dimensional) Borel probability measures over the push-forward measure
If we equip with the product topology of weak convergence on each and the topology of convergence of the finite dimensional distributions on and , then the mapping given by
is continuous and surjective. In particular, Sklar’s theorem holds.
(1) Since products of Hausdorff spaces are Hausdorff and is homeomorphic to a subset of a product of Hausdorff spaces, it is Hausdorff.
(2) We know by [DFS12, Thm. 3.3] that every is compact with respect to the topology of weak convergence on . Tychonoff’s Theorem guarantees also that is compact with respect to the product topology on . Therefore, as is closed by Lemma 2.1, we obtain that is compact, since it is homeomorphic to an intersection of a closed and a compact set in the product topology.
is a -nullset for all , , since then we immediately obtain
(4) Define by
which is well defined by (3). Since the finite-dimensional distributions of a law are consistent, forms a compatible family. Define analogously for also by
This is by Sklar’s theorem in finite-dimensions surjective and by [SEM04, Thm. 2] also continuous. Hence is continuous and surjective, since both, and are. must be the uniquely induced continuous mapping by the family by the universality property of the inverse limit. Moreover, since by [SS05, Corollary 4.2.6] is compact and by (2) also is compact, we have that is compact by Tychonoff’s theorem. The continuity of implies therefore that is compact, hence closed. Since moreover Lemma 2.2 implies that is dense, we obtain that is surjective and therefore also the first part of Sklar’s theorem holds. The uniqueness of the copulas in the case of continuous marginals follows immediately by Sklar’s theorem in finite dimensions via the uniqueness of the finite dimensional distribution of the corresponding copula measure. ∎
Observe that since is a locally convex Hausdorff space with respect to the topology of weak convergence for each , we obtain that also the inverse limit is locally convex, as it is isomorphic to a subset of the product of locally convex Hausdorff spaces. Hence, as mentioned for instance in [DS15, p.30], since is convex, we have that it is the closure of its extremal points by the Krein-Milman theorem. As mentioned in [BS91] this implies that
where denotes the set of extremal points of and is a convex function.
This research was funded within the project STORM: Stochastics for Time-Space Risk Models, from the Research Council of Norway (RCN). Project number: 274410.
Extremal solutions in the marginal problem.
Advances in Probability Distributions with Given Marginals. Mathematics and Its Applications, G. Dall’Aglio, S. Kotz, and G. Salinetti (Eds.), Vol. 67. Cited by: §4.
- [BDS20] (2020) Copula measures and sklar’s theorem in arbitrary dimensions. E-print arXiv:2012.11530. Cited by: §1, §3, §3, §4.
- [CT02] (2002) A new proof of Sklar’s theorem. In Distributions With Given Marginals and Statistical Modelling, C. M. Cuadras, J. Fortiana, and J. A. Rodriguez-Lallena (Eds.), pp. 29–34. Cited by: §1.
- [DEH09] (2009) A multivariate bahadur–kiefer representation for the empirical copula process. J. Math. Sci. (N. Y.) 163, pp. 382–398. Cited by: §1.
- [DFS12] (2012) Sklar’s theorem obtained via regularization techniques. Nonlinear Anal. 75 (2), pp. 769–774. Cited by: §1, §1, §4.
- [DFS13] (2013) A topological proof of Sklar’s theorem. Appl. Math. Lett. 26 (9), pp. 945–948. Cited by: §1, §3.
- [DS15] (2015) Principles of copula theory. CRC/Chapman & Hall, London. Cited by: §4.
- [MS75] (1975) Unified large-sample theory of general chi-squared statistics for tests of fit. Ann. Statist. 3, pp. 599–616. Cited by: §1.
- [RZ10] (2010) Profinite Groups. Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 40, Springer-Verlag. Cited by: §2, §2, §2.
- [RÜS09] (2009) On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139 (11), pp. 3921–3927. Cited by: §1, §3.
Operations on distribution functions not derivable from operations on random variables. Studia Math. 52, pp. 43–52. Cited by: §1.
- [SS05] (2005) Probabilistic Metric Spaces. Dover Publications, New York. Cited by: §4, §4.
- [SEM04] (2004) Convergence of copulas: critical remarks. Rad. Mat. 12 (2), pp. 241–249. Cited by: §4.
- [SKL96] (1996) Random variables, distribution functions, and copulas—a personal look backward and forward. In Distributions with fixed marginals and related topics, L. Rüschendorf, B. Schweizer, and M. D. Taylor (Eds.), Lecture Notes–Monograph Series, Vol. Volume 28, pp. 1–14. Cited by: §1.