The Randomness Deficiency Function and the Shift Operator

TL;DR

This paper investigates how the measure of randomness deficiency between two infinite sequences behaves under repeated shifts, showing that their difference can grow without bound.

Contribution

It introduces the concept that the difference in randomness deficiencies can become unbounded through shift iterations, advancing understanding of sequence randomness.

Findings

Difference in randomness deficiencies can be unbounded

Shift operator impacts sequence randomness measures

Almost sure divergence under repeated shifts

Abstract

Almost surely, the difference between the randomness deficiencies of two infinite sequences will be unbounded with respect to repeated iterations of the shift operator.