On a class of norms generated by nonnegative integrable distributions

Michael Falk 1  and Gilles Stupfler 2
  • 1 Institute of Mathematics, University of Würzburg, Würzburg, Germany
  • 2 School of Mathematical Sciences, University of Nottingham, Nottingham

Abstract

We show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F-norm can easily be estimated by an empirical F-norm, whose consistency and weak convergence we establish.

The concept of F-norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of F-norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of F-norms.

We conclude by showing how, using the geometry of F-norms, we may characterize nonnegative integrable distributions in ℝd by simple compact sets in ℝd+1. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.

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Dependence Modeling aims to provide a medium for exchanging results and ideas in the area of multivariate dependence modeling. Topics include Copula methods, environmental sciences, estimation and goodness-of-fit tests, extreme-value theory, limit laws, mass transportations, measures of association, multivariate distributions and tests, quantitative risk management, risk assessment, risk models, risk measures and stochastic orders and time series.

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