# Effective Aspects of Bernoulli Randomness

## TL;DR

This paper investigates Bernoulli random sequences, especially when the Bernoulli parameter is noncomputable, revealing properties of their randomness, Turing degrees, and limitations of blind Bernoulli randomness characterizations.

## Contribution

It establishes that proper Bernoulli parameters must be proper if a sequence is both proper and Bernoulli random, and analyzes the Turing degrees and characterization challenges of Bernoulli randomness.

## Key findings

- Proper Bernoulli parameters are necessary for proper Bernoulli random sequences.
- Turing degrees of Bernoulli random sequences differ from those of Martin-Löf random sequences.
- Blind Bernoulli randomness cannot be fully characterized by tests lacking access to parameter p.

## Abstract

In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-L\"of random with respect to a Bernoulli measure $\mu_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on the case in which the underlying Bernoulli parameter $p$ is proper (that is, Martin-L\"of random with respect to some computable measure). We show for every Bernoulli parameter $p$, if there is a sequence that is both proper and Martin-L\"of random with respect to $\mu_p$, then $p$ itself must be proper, and explore further consequences of this result. We also study the Turing degrees of Bernoulli random sequences, showing, for instance, that the Turing degrees containing a Bernoulli random sequence do not coincide with the Turing degrees containing a Martin-L\"of random sequence. Lastly, we consider several possible approaches to characterizing blind Bernoulli randomness, where the corresponding Martin-L\"of tests do not have access to the Bernoulli parameter $p$, and show that these fail to characterize blind Bernoulli randomness.