Algorithmic randomness and stochastic selection function

TL;DR

This paper extends classical theorems on normal numbers to the realm of algorithmic randomness, characterizing subsequences that preserve randomness properties using ML-random sequences and complexity rates.

Contribution

It provides the first algorithmic randomness versions of Kamae-Weiss and Steinhaus theorems, connecting classical normal number theory with algorithmic complexity.

Findings

Characterizes selection functions preserving ML-randomness.

Establishes conditions for subsequences to maintain normality in algorithmic terms.

Bridges classical normal number theorems with algorithmic randomness theory.

Abstract

We show algorithmic randomness versions of the two classical theorems on subsequences of normal numbers. One is Kamae-Weiss theorem (Kamae 1973) on normal numbers, which characterize the selection function that preserves normal numbers. Another one is the Steinhaus (1922) theorem on normal numbers, which characterize the normality from their subsequences. In van Lambalgen (1987), an algorithmic analogy to Kamae-Weiss theorem is conjectured in terms of algorithmic randomness and complexity. In this paper we consider two types of algorithmic random sequence; one is ML-random sequences and the other one is the set of sequences that have maximal complexity rate. Then we show algorithmic randomness versions of corresponding theorems to the above classical results.