Mutual Dimension and Random Sequences

If S and T are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions mdim(S : T ) and Mdim(S : T ) are the upper and lower densities of the algorithmic information that is shared by S and T . In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences R1 and R2 with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions mdim(R1 : R2) and Mdim(R1 : R2), and we show that the condition Mdim(R1 : R2) = 0 is necessary but not sufficient for R1 and R2 to be independently random. We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S : T ) and Mdim(S : T ) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the “correct” relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T ), and Dim(T ) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.