On Asymptotic Proximity of Distributions

TL;DR

This paper investigates the convergence of probability measure sequences in metric spaces without assuming tightness, comparing various definitions and establishing general properties of such convergence.

Contribution

It provides a comprehensive analysis of convergence concepts for probability measures without the tightness assumption, filling a gap in the theoretical understanding.

Findings

Different definitions of convergence are compared.

General properties of non-tight convergence are established.

The work broadens the theoretical framework for probability measure convergence.

Abstract

We consider some general facts concerning convergence P_{n}-Q_{n}\to 0 as n\to \infty, where P_{n} and Q_{n} are probability measures in a complete separable metric space. The main point is that the sequences {P_{n}} and {Q_{n}} are not assumed to be tight. We compare different possible definitions of the above convergence, and establish some general properties.