Effective weak and vague convergence of measures on the real line

TL;DR

This paper develops an effective framework for understanding weak and vague convergence of measures on the real line, establishing equivalences, computability conditions, and extending classical results into an effective setting.

Contribution

It introduces an effective theory for weak and vague convergence, proves their equivalence under certain conditions, and explores computability aspects of limits in this framework.

Findings

Effective convergence in the Prokhorov metric is equivalent to effective weak convergence.

Both uniform and non-uniform notions of effective vague convergence are equivalent.

Conditions are provided for when effective vague limits are computable.

Abstract

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.