Minimum correlation for any bivariate Geometric distribution

Abstract

Consider a bivariate Geometric random variable where the first component has parameter p 1 and the second parameter p 2 . 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 timeOne method for generating a bivariate geometric with target correlation requires computing this minimum correlation. The minimum correlation is shown to be nonmonotonic in p 1 and p 2 , moreover, the partial derivatives are not continuous. For p 1 = p 2 , 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.