Relations between randomness deficiencies

TL;DR

This paper explores the relationships between various definitions of randomness deficiencies, demonstrating that differences among them can be arbitrarily large, which impacts understanding of randomness in sequences.

Contribution

It shows that the differences between certain randomness deficiency functions can be unbounded, clarifying the limits of their equivalence.

Findings

Differences between some deficiency functions can be arbitrarily large.

All deficiency functions are bounded by a logarithmic term.

The results clarify the relationships among various randomness measures.

Abstract

The notion of random sequence was introduced by Martin-Loef in 1966. At the same time he defined the so-called randomness deficiency function that shows how close are random sequences to non-random (in some natural sense). Other deficiency functions can be obtained from the Levin-Schnorr theorem, that describes randomness in terms of Kolmogorov complexity. The difference between all of these deficiencies is bounded by a logarithmic term. In this paper we show that the difference between some deficiencies can be as large as possible.