The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some specified distance $r$.
Let $x \sim N(\mu_1, \Sigma_1),$ where $\mu_1$ is a length $3M$ vector and $\Sigma_1$ is a $3M \times 3M$ positive definite matrix.
Let $y \sim N(\mu_2, \Sigma_2),$ where $\mu_2$ is a length $3N$ vector and $\Sigma_2$ is a $3N \times 3N$ positive definite matrix.
I want to know an approximation for a given $r$ to the probability that for all $i<M$ and $j<N$, $(x_{3i+0}-y_{3j+0})^2 + (x_{3i+1}-y_{3j+1})^2 + (x_{3i+2}-y_{3j+2})^2 > r^2$.
Both $M$ and $N$ are between 5 and 10. I need to solve many such problems, so I am more interested in a rough approximation that is very fast than a slow but accurate solution.
It is safe to assume that the diagonal elements of the $\Sigma_1$ and $\Sigma_2$ matrices are much larger than the off-diagonal elements, if that makes the problem easier. Still, I do not want to set the off-diagonal elements of the correlation matrices to exactly zero.
Thank you.
