Bernoulli randomness and Bernoulli normality

Abstract One can consider μ‐Martin‐Löf randomness for a probability measure μ on 2ω, such as the Bernoulli measure given . We study Bernoulli randomness of sequences in with parameters , and we reintroduce Bernoulli normality, where the uniform distribution of digits is replaced with a Bernoulli distribution. We prove the equivalence of three characterizations of Bernoulli normality. We show that every Bernoulli random real is Bernoulli normal, and this has the corollary that the set of Bernoulli normal reals has full Bernoulli measure in . We give an algorithm for computing Bernoulli normal sequences from normal sequences so that we can give explicit examples of Bernoulli normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and Bernoulli normality.