Suppose there are $n$ jointly normal random variables with positive correlation $t_i$ where $i=1,..,n$. We are tasked with calculating $P(\sum_{i=1}^{n} t_i \leq T)$ by separating the sum in two parts: from $1$ to $m$ and $m+1$ to $n$.
What I am trying to do is the following:
$P(\sum_{i=1}^{n} t_i \leq T) = P(\sum_{i=1}^{m} t_i \leq T) + P(\sum_{i=m+1}^{n} t_i \leq T) $
However this is not possible because of the correlation.
Is there a way to obtain some type of bound or a way to calculate the probability from the sum of the partial sums?