Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certainclasses of sub-computable functions which lack a universal machine. Inparticular, we define such versions of randomness for primitive recursivefunctions and for PSPACE functions. These new notions are robust in that thereare equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorovcomplexity, and (3) martingales. We show these notions can be equivalentlydefined with prefix-free Kolmogorov complexity. We prove that one direction ofvan Lambalgen's theorem holds for relative computability, but the otherdirection fails. We discuss statistical properties of these notions ofrandomness