Seminar

When coupling discretely "copulates"

Olivier Faugeras (University Toulouse 1 - Capitole)

March 18, 2014, 14:00–15:15

Toulouse

Room MF 323

Statistics Seminar

Abstract

Coupling is a powerful probabilistic method which allows to turn distributional properties into their corresponding counterparts in terms of random variables. Skorohod's theorem, stochastic orders, recurrence and convergence to equilibrium of Markov chains (Doeblin's method), renewal and regeneration theory can often be treated in an elegant and powerful way via coupling techniques. Copula functions allows to express a multivariate cdf in terms of its marginal cdfs and are useful to model dependence. We intent to show in this speech that the proximity of copulas and coupling methods is not only semantic but also fruitful from a mathematical point of view. Here, we focus on copulas associated to a discrete vector, where delicate issues arises, due to the non-unicity of the corresponding copulas. More precisely, we use a maximal coupling construction to show a strenghtened form of Skorohod's theorem applied to the empirical cdf of a purely discrete vector X: for an i.i.d. or even ergodic sample, one can construct, on a common probability space, a sequence of random variables, distributed according to the ecdf, which not only converges but remains constant and equal to some copy of X, after some a.s. finite random time. This preliminary result is applied to two versions of genuine empirical copulas obtained either via probabilistic continuation, i.e. kernel smoothing, or via the distributional transform, to give simple short proofs of uniform a.s. convergence of empirical copulas on their whole domain. The former result extends some recent results of Faugeras [2013] JMVA, while the latter is new and gives a positive answer to the questions raised by Lindner, Szimayer and Neshelova on the convergence of copulas in the discrete case. Moreover, it gives, as a simple corollary, consistency of recent proposals of extensions of concordance measures to discrete vectors, such as Neshelova's JMVA [2007] and Mesfioui and Quessy's JMVA [2010] extensions of Kendall's tau.