Weak uniform structures on probability distributions

TL;DR

This paper explores uniform structures on probability distributions that generalize the weak topology, showing they lead to consistent asymptotic approximation for sequences of distributions.

Contribution

It introduces a class of uniform structures on probability distributions that extend the weak topology and demonstrates their equivalence for sequences in asymptotic analysis.

Findings

All structures in the class produce the same asymptotic approximation for sequences.

The structures generalize the weak topology to broader uniform structures.

Consistency holds for sequences, not necessarily for general nets.

Abstract

In dealing with asymptotic approximation of possibly divergent nets of probability distributions, we are led to study uniform structures on the set of distributions. This paper identifies a class of such uniform structures that may be considered to be reasonable generalizations of the weak topology. It is shown that all structures in the class yield the same notion of asymptotic approximation for sequences (but not for general nets) of probability distributions.