Minimum Correlation for Any Bivariate Geometric Distribution

Consider a bivariate Geometric random variable where the first component has parameter p1 and the second parameter p2. It is not possible to make the correlation between the marginals equal to -1. Here the properties of this minimum correlation are studied both numerically and analytically. It is shown that the minimum correlation can be computed exactly in time O(p −1 1 ln(p −1 2 ) + p −1 2 ln(p −1 1 )). One method for generating a bivariate geometric with target correlation requires computing this minimum correlation. The minimum correlation is shown to be nonmonotonic in p1 and p2, moreover, the partial derivatives are not continuous. For p1 = p2, these discontinuities are characterized completely and shown to lie near (1 - roots of 1/2). In addition, we construct analytical bounds on the minimum correlation.