An effectivization of the law of large numbers for algorithmically random sequences and its absolute speed limit of convergence

TL;DR

This paper demonstrates that the law of large numbers can be effectively applied to Schnorr random sequences with computable measures, establishing a universal speed limit of convergence of 2, and extends these results to broader probabilistic contexts.

Contribution

It introduces an effective version of the law of large numbers for Schnorr random sequences and identifies a universal convergence speed limit of 2, extending to non-computable probability spaces.

Findings

Law of large numbers is effectivized for Schnorr random sequences.

A universal speed limit of convergence of 2 is established.

Effectivization of almost sure convergence is provided for broader probability spaces.

Abstract

The law of large numbers is one of the fundamental properties which algorithmically random infinite sequences ought to satisfy. In this paper, we show that the law of large numbers can be effectivized for an arbitrary Schnorr random infinite sequence, with respect to an arbitrary computable Bernoulli measure. Moreover, we show that an absolute speed limit of convergence exists in this effectivization, and it equals 2 in a certain sense. In the paper, we also provide the corresponding effectivization of almost sure convergence in the strong law of large numbers, and its absolute speed limit of convergence, in the context of probability theory, with respect to a large class of probability spaces and i.i.d. random variables on them, which are not necessarily computable.