Schnorr randomness for noncomputable measures

Schnorr randomness for noncomputable measures

This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say x is uniformly Schnorr μ-random if t(μ, x) < ∞ for all lower semicomputable functions t(μ, x) such that μ 7→ ∫ t(μ, x) dμ(x) is computable. We prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as MartinLöf randomness for noncomputabl...

Schnorr randomness

Subcomputable Schnorr Randomness

The notion of Schnorr randomness refers to computable reals or computable functions. We propose a version of Schnorr randomness for subcomputable classes and characterize it in different ways: by Martin-Löf tests, martingales or measure computable machines. Mathematics Subject Classification: 03D25, 68Q15 .

Schnorr Randomness and the Lebesgue Differentiation Theorem

We exhibit a close correspondence between L1-computable functions and Schnorr tests. Using this correspondence, we prove that a point x ∈ [0, 1] is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at x for all L1-computable functions f ∈ L1([0, 1]).

On Schnorr and computable randomness, martingales, and machines

This paper falls within an overall program articulated in Downey, Hirschfeldt, Nies and Terwijn [8], and Downey and Hirschfeldt [4], of trying to calibrate the algorithmic randomness of reals. There are three basic approaches to algorithmic randomness. They are to characterize randomness in terms of algorithmic predictability (“a random real should have bits that are hard to predict”), algorith...