The Smallest Probability Interval a Sequence Is Random for: A Study for Six Types of Randomness

TL;DR

This paper investigates the smallest probability intervals for which a sequence is considered random across six different randomness notions, revealing how these intervals compare and under what conditions they coincide or differ.

Contribution

It introduces the concept of smallest probability intervals for various randomness notions and analyzes their properties and relationships.

Findings

Every sequence has a smallest interval it is (almost) random for under many notions.

Conditions are established for when these smallest intervals coincide.

Examples are provided where the smallest intervals differ across notions.

Abstract

There are many randomness notions. On the classical account, many of them are about whether a given infinite binary sequence is random for some given probability. If so, this probability turns out to be the same for all these notions, so comparing them amounts to finding out for which of them a given sequence is random. This changes completely when we consider randomness with respect to probability intervals, because here, a sequence is always random for at least one interval, so the question is not if, but rather for which intervals, a sequence is random. We show that for many randomness notions, every sequence has a smallest interval it is (almost) random for. We study such smallest intervals and use them to compare a number of randomness notions. We establish conditions under which such smallest intervals coincide, and provide examples where they do not.