Effective notions of weak convergence of measures on the real line

TL;DR

This paper develops an effective framework for understanding weak convergence of measures on the real line, establishing the equivalence of two notions and providing an effective version of the Portmanteau Theorem.

Contribution

It introduces two effective notions of weak convergence on , proves their equivalence, and derives an effective Portmanteau Theorem.

Findings

Two effective notions of weak convergence are equivalent.

An effective version of the Portmanteau Theorem is established.

The framework facilitates computational approaches to measure convergence.

Abstract

We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on $\mathbb{R}$: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.