The interest in this talk is in statistical (in)dependence between
a finite number of random vectors. Statistical independence between
random vectors holds if and only if the true underlying copula is the product
of the marginal copulas yielding zero dependence.
We discuss some recent approaches towards developing dependence measures
that completely characterize independence, such as
phi-divergence measures, and optimal transport measures.
We discuss statistical inference properties and provide illustrative examples.
In high-dimensional settings possible marginal independencies can be taken
into account by inducing (block) sparsity.

We examine the bivariate copulas that describe the serial dependencies in popular time series models from the ARCH/GARCH class. We show how these copulas can be approximated using a combination of standard bivariate copulas and uniformity-preserving transformations known as v-transforms. The insights can help us to construct stationary d-vine models to rival and often surpass the performance of GARCH processes in modelling and forecasting volatile financial return series.

We study the large-sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their empirical counterparts. It is well known that the latter modification significantly alters the limiting laws compared to usual M-estimation. We establish the consistency and the asymptotic normality of our sparse penalized M-estimator and we prove the asymptotic oracle property with pseudo-observations, possibly in the case when the number of parameters is diverging. Our framework allows to manage copula-based loss functions that are potentially unbounded. Additionally, we state the weak limit of multivariate rank statistics for an arbitrary dimension and the weak convergence of empirical copula processes indexed by maps.
We apply our inference method to Canonical Maximum Likelihood losses with Gaussian copulas, mixtures of copulas or conditional copulas. The theoretical results are illustrated by two numerical experiments.