Computability of Probability Distributions and Characteristic Functions

TL;DR

This paper explores the computability of probability distributions and their characteristic functions, establishing effective versions of classical theorems like Glivenko's, Bochner's, and the central limit theorem.

Contribution

It introduces effective versions of key theorems in probability theory, linking computability of distributions to their characteristic functions.

Findings

A sequence of distributions is computable iff their characteristic functions are computable.

Effective Glivenko's theorem is established.

Effectivizations of Bochner's theorem and the central limit theorem are proved.

Abstract

As a part of our works on effective properties of probability distributions, we deal with the corresponding characteristic functions. A sequence of probability distributions is computable if and only if the corresponding sequence of characteristic functions is computable. As for the onvergence problem, the effectivized Glivenko's theorem holds. Effectivizations of Bochner's theorem and de Moivre-Laplace central limit theorem are also proved.