On a class of norms generated by nonnegative integrable distributions

TL;DR

This paper introduces $F$-norms generated by nonnegative integrable distributions, characterizes their properties, and explores their convergence, estimation, and geometric representation, linking distribution convergence to Hausdorff distances.

Contribution

It defines and characterizes $F$-norms from nonnegative integrable distributions, establishes their convergence and estimation properties, and connects distribution convergence to geometric sets.

Findings

$F$-norms uniquely characterize distributions with nonnegative integrable marginals.

Pointwise convergence of $F$-norms is equivalent to Wasserstein convergence of distributions.

$F$-norms can be estimated consistently and weakly converged in statistical applications.

Abstract

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