  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 $R_1$ and
$R_2$ 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 $mdim(R_1:R_2)$ and
$Mdim(R_1:R_2)$, and we show that the condition $Mdim(R_1:R_2) = 0$ is
necessary but not sufficient for $R_1$ and $R_2$ 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.
