Bernoulli Randomness and Biased Normality

TL;DR

This paper explores Bernoulli randomness and introduces a biased version of normality, proving that Bernoulli random sequences are biased normal, and provides algorithms and applications related to these concepts.

Contribution

It introduces biased normality, proves Bernoulli randomness implies biased normality, and offers algorithms for constructing biased normal sequences from normal sequences.

Findings

Bernoulli random reals are biased normal.

Biased normal reals have full Bernoulli measure.

Algorithms enable explicit biased normal sequence construction.

Abstract

One can consider $\mu$-Martin-L\"of randomness for a probability measure $\mu$ on $2^{\omega}$, such as the Bernoulli measure $\mu_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^{\omega}$ with parameters $p_0, p_1, \dotsc, p_{n-1}$, and we introduce a biased version of normality. We prove that every Bernoulli random real is normal in the biased sense, and this has the corollary that the set of biased normal reals has full Bernoulli measure in $n^{\omega}$. We give an algorithm for computing biased normal sequences from normal sequences, so that we can give explicit examples of biased normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and biased normality.