Minimum correlation for any bivariate geometric distribution (Q2921125)

The aim of this paper is to investigate the minimum attainable correlation between two geometric random variables. It is shown that \(\min\{\text{corr}(x_1,x_2):x_1\sim\text{Geo}\,p_1, x_2\sim\text{Geo}\, p_2\}=\rho_-(p_1,p_2)\) can be computed exactly in time O\((p^{-1}_1\ln p^{-1}_2+p_2\ln p^{-1}_1)\). The minimum correlation is shown to be nonmonotonic in \(p_1\) and \(p_2\) and the partial derivarives are not continuous. Moreover, the points where the discontinuities occur are near to (1-roots of 1/2). Finally, the authors construct analytic bounds on the minimum correlation.