We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows:
ΩU[X] is random whenever X is Σ⁰n-complete or Π⁰n-complete for some n≥2.
However, for n≥2, ΩU[X] is not n-random when X is Σ⁰n or Π⁰n. Nevertheless, there exists Δ⁰n+1 sets such that ΩU[X] is n-random.
There are Δ⁰₂ sets X such that ΩU[X] is rational. Also, for every n≥1, there exists a set X which is Δ⁰n+1 and Σ⁰n-hard such that ΩU[X] is not random.
The same questions are also considered in the context of infinite computations, and lead to similar results.
"Randomness and halting probabilities." J. Symbolic Logic 71 (4) 1411 - 1430, December 2006. https://doi.org/10.2178/jsl/1164060463