# Multivariate T distribution?

Suppose $$(Y_1,Y_2)^T \sim N((0,0)^T,\Sigma)$$, where $$\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}.$$ $$T_1= \frac{\bar{Y_1}}{S_1/\sqrt(1/n)}$$ and $$T_2= \frac{\bar{Y_2}}{S_2/\sqrt(1/n)}$$ are the T test statistics. If $$\sigma_1 = \sigma_2$$ and $$S_1=S_2$$, then the joint distribution of $$T_1, T_2$$ is multivariate t distribution with the shape matrix equal to the correlation matrix corresponding to $$\Sigma$$ and $$df=2n-2$$. What is the joint distribution of $$T_1, T_2$$ if $$\sigma_1 \ne \sigma_2$$ ? Can I estimate it by the multivariate t distribution with shape matrix equal to the correlation matrix corresponding to $$\Sigma$$ and $$df=n-1$$ ?