We study a distributed sampling problem where set of processors want to output (approximately) independent and identically samples from given joint distribution with the help common message coordinator. Each processor has access subset sources “shared” randomness. consider two cases - in “omniscient coordinator setting”, all these shared randomness, while “oblivious setting,” it none. In additi...

We study the strengths of various notions of higher randomness: (i) strong Π1-ML-randomness is separated from Π 1 1-ML-randomness; (ii) the hyperdegrees of Π1-random reals are closed downwards (except for the trivial degree); (iii) the reals z in NCRΠ11 are precisely those satisfying z ∈ Lωz 1 ; and (iv) lowness for ∆1-randomness is strictly weaker than that for Π 1 1-randomness.