A d-dimensional positive definite correlation matrix R=([rho]ij) can be parametrized in terms of the correlations [rho]i,i+1 for i=1,...,d-1, and the partial correlations [rho]iji+1,...j-1 for j-i[greater-or-equal, slanted]2. These parameters can independently take values in the interval (-1,1). Hence we can generate a random positive definite correlation matrix by choosing independent distributions Fij, 1[less-than-or-equals, slant]i<j[less-than-or-equals, slant]d, for these parameters. We obtain conditions on the Fij so that the joint density of ([rho]ij) is proportional to a power of det(R) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in -dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det(R) as a function of [rho]i,i+1 for i=1,...,d-1, and [rho]iji+1,...j-1 for j-i[greater-or-equal, slanted]2.
