Sam Buss, Douglas Cenzer and Jeffrey B. Remmel
    "Sub-computable Bounded Randomness"
    Logical Methods of Computer Science 10, 4 (2014) paper 15.

    Download manuscript: PDF.

Abstract: This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-Lof tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.

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