Tail asymptotics for the bivariate skew normal in the general case

The present paper is a sequel to and generalization of Fung and Seneta (2016) whose main result gives the asymptotic behaviour as $ u \to 0^{+}$ of $\lambda_L(u) = P(X_1 \leq F_1^{-1}(u) | X_2 \leq F_2^{-1}(u)),$ when $\bf{X} \sim SN_2(\boldsymbol{\alpha}, R)$ 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 $\textbf{X}$. A paper of Beranger et al. (2017) enunciates an upper-tail version which does not contain the constraint $\alpha_1=\alpha_2= \alpha$ but requires the constraint $0 <\rho <1$ in particular. The proof, in their Appendix A.3, is very condensed. When translated to the lower tail setting of Fung and Seneta (2016), we find that when $\alpha_1=\alpha_2= \alpha$ the exponents of $u$ in the regularly varying function asymptotic expressions do agree, but the slowly varying components, always of asymptotic form $const (-\log u)^{\tau}$, are not asymptotically equivalent. Our general approach encompasses the case $ -1 <\rho < 0$, and covers all possibilities.