Is there any way of approximating or analytically solving the below CDF (let's say even for $n\to\infty$)?
I am trying to find the below probability: \begin{align} &P\left[X_{2}-X_{1} \leq 0,X_{3}-X_{1} \leq 0, X_{4}-X_{1} \leq 0 \right]\\ &= F_{\tilde{X_{2}},\tilde{X_{3}},\tilde{X_{4}}}(0,0,0) \end{align}
Where we know all the distributions (normal with known mean and variance) and we also know that the random variables are correlated.This method of solving seems very tedious and does not generalize well if we have even more random variables.
Hint:
Find an $3\times 4$ matrix $A$ such that $$A(X_1,X_2,X_3,X_4)^T=(X_2-X_1,X_3-X_1,X_2-X_1)^T=(\tilde{X_2},\tilde{X_3},\tilde{X_4})^T$$
If $(X_1,X_2,X_3,X_4)^T$ has normal distribution with expectation $\mu$ and covariance $\Sigma$ then $(\tilde{X_2},\tilde{X_3},\tilde{X_4})^T$ has normal distribution with expectation $A\mu$ and covariance matrix $A\Sigma A^T$.
Working with this new distribution will probably make the calculation more simple.