Propagation of partial randomness

Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong f -randomness implies strong f randomness relative to a PA-degree. We also prove: if X is strongly f -random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f -random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f -randomness relative to a PA-degree implies strong f -randomness, but f -randomness does not imply f -randomness relative to a PA-degree.