The present paper is a sequel to and generalization of [10] whose main result gives the asymptotic behaviour as \( u \rightarrow 0^{+}\) of \(\lambda _L(u) = P(X_1 \le F_1^{-1}(u) | X_2 \le F_2^{-1}(u)),\) when \({\textbf {X}} \sim SN_2(\varvec{\alpha }, R)\) with \(\alpha _1 = \alpha _2 = \alpha \) , that is: for the bivariate skew normal distribution in the equi-skew case, where R is the correlation matrix, with off-diagonal entries \(\rho \) , and \(F_i(x)\) , \(i=1,2\) are the marginal cdf’s of \({\textbf {X}}\) . In the present paper we show in full generality that \( \lambda _L(u) \sim Ku^{\theta }(-\log u)^{\tau }\) , and find explicit expressions for K, \(\theta \) , \(\tau \) in all possible settings, without any restriction on \(\rho , -1< \rho <1.\) Our analysis is copula based. A paper of [7] enunciates a general upper-tail version, when \(0< \rho <1.\) When translated to the lower tail setting of [10], we find that in the case \(\alpha _1=\alpha _2= \alpha \) only the exponents, \(\theta \) , of u in the regularly varying function asymptotic expressions of [7, 10] do agree, but the slowly varying components are not asymptotically equivalent.
