On a class of norms generated by nonnegative integrable distributions

Gilles Stupfler,Michael Falk

doi:10.1515/demo-2019-0014

Gilles Stupfler, Michael Falk

Open Access

https://doi.org/10.1515/demo-2019-0014

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Abstract

Abstract We show that any distribution function on ℝ d with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝ d+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 ℝ d by simple compact sets in ℝ d+1. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.

Highlights

It was observed only recently that a particular kind of norms on Rd, called D-norms, are the skeleton of multivariate extreme value theory
D-norms are tailor-made for multivariate extreme value theory as they turn out to provide an accessible common thread, in the sense that they do not require the knowledge of multivariate regular variation and of the associated topology background
Our paper introduces F-norms, which are an o spring of D-norms, to address the general framework of multivariate distributions rather than the max-stable distributions that are the focus of multivariate extreme value theory

Summary

Introduction

It was observed only recently that a particular kind of norms on Rd, called D-norms, are the skeleton of multivariate extreme value theory. The distribution function (df) of this rv, is not uniquely determined, and there exists an in nite number of generators of the same D-norm. It was shown by [6] that the D-norm characterizes the distribution of a generator if the constant function one is added to the generator as a further component. This led to the de nition of the max-characteristic function, which can be used to identify the distribution of any multivariate distribution with nonnegative and integrable components

Objectives

Conclusion