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Stochastic Analysis
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Schroers, Dennis (2020). Copulas and Sklar’s Theorem in Infinite Dimensions.
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Copulas describe statistical dependence between the components of multivariate random variables in full generality by virtue of Sklar’s theorem. Although they are used and defined for certain infinite dimensional objects (e.g. Gaussian processes in [WG10], Markov processes in [DNO92], [Ibr09] and [L10], or copulas in Hilbert spaces in [HR17]) there is no prevalent notion of a copula as an infinite dimensional law that unifies these concepts. To this end we define copulas as probability measures on product spaces RI for some index set I and formulate Sklar’s theorem in this general setting. An important application is the task of modeling the inherent dependence of random variables in function spaces. In Banach spaces (or more general topological vector spaces) already the notion of marginals becomes ambiguous: Take the example of the space C[0,1]. it is possible to emded this space into the product space R[0,1] of real functions on [0,1] via (1) f ↦ ( f (x))x∈[0,1] At the same time, the existence of a Schauder basis in this space makes it possible to embded it into the space RN of real sequences via
(2) f ↦ (en( f ))n∈N for the coefficient functionals (en)n∈N of the Schauder basis. Both embeddings lead to reason able notions of marginals of a measure μ on C[0,1], either as the laws (μ(δ−1⋅) for the −1 x x∈[0,1] evaluation functionals δxh = h(x), as well as (μ(en ⋅)n∈N. For this reason, copula on function spaces have to be considered for each of such a proper dual subset separately, depending on the respective modeling task. In finite dimensions the second part of Sklar’s Theorem enables us to construct probability measures by merging arbitrary marginals and dependence structures (in form of copulas). This simple construction cannot be transferred into the infinite dimensional setting of function spaces without any problems: Taking a set of marginals (μ(δ−1⋅) in terms of the function x x∈[0,1] embedding (1) on C[0,1], and a set of absolutely continuous marginal distributions (Ft)t∈[0,1], we need these marginals to be continuous in distribution, that is t ↦ Ft(x) has to be continuous for each x ∈ R. This is clearly not the case for every choice of marginals and the constructed measure would rather be a cylindrical premeasure on C[0,1]. We will therefore provide criteria to decide if the construction induced by the second part of Sklar’s theorem yields certain properties like the regularity of paths and culminates in a real probability law in a function space. In addition, following [CARTD93] and [AJ14], we link our concept of copulas to Wasserstein distances. Namely, underlying copulas of probability laws in lp sequence spaces with finite pth moment effectively solve a marginalrestricted optimisation problems in the pth Wasserstein space.

Schroers, Dennis (2020). Copulas and Sklar's Theorem in Infinite Dimensions.
Show summary
Copulas describe statistical dependence between the components of multivariate random variables in full generality by virtue of Sklar’s theorem. Although they are used and defined for certain infinite dimensional objects (e.g. Gaussian processes, Markov processes or infinite dimensional Archimedean copulas) there is no prevalent notion of a copula as an infinite dimensional law that unifies these concepts. To this end we define copulas as probability measures on product spaces and prove Sklar’s theorem in this general setting. Afterwards we use this result on Banach spaces to construct cylindrical probability measures with predefined marginals and underlying copula. This induces the functional analytic problem of finding criteria in which cases the obtained cylindrical law induces a real probability measure, which is in general difficult to decide. We solve this problem in the pWasserstein space on the space of psummable sequences (including separable Hilbert spaces) and show that copulas effectively solve a restricted optimal coupling problem.

Schroers, Dennis (2020). Sklar's Theorem in Infinite Dimensions.
Published Nov. 13, 2019 11:39 AM
 Last modified Nov. 14, 2019 8:42 AM