I would like to compute the following summation:
$$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$
where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real numbers, and $u_i$ are in $[0, 1]$ so that $u_i = 0$ for some $i$s, $u_i = 1$ for some other $i$s, and $u_i \in (0, 1)$ for the rest. In other words, the sum has several finite elements, several elements equal to $+\infty$, and several elements equal to $-\infty$. As far as I understand, the result is undefined in general; however, we know what those infinities are.
My idea is to break the sum into three sums as follows:
$$ \begin{align*} s &= \sum_{i} a_i \, \Phi^{-1}(u_i) + \Phi^{-1}(1) \sum_{i} a_i + \Phi^{-1}(0) \sum_{i} a_i\\ &= s_1 + \Phi^{-1}(1) s_2 + \Phi^{-1}(0) s_3. \end{align*} $$
Then I am planning to proceed as follows. If $s_1 = s_2$, the sum is set to $s_1$. (This also includes the zero case, which is already arguable.) Depending on the signs and magnitudes of $s_1$ and $s_2$, the sum is set to $+\infty$ or $-\infty$.
My question is: Does the above procedure make sense? Is it legitimate to operate with infinities as I described?
I would appreciate any thoughts and suggestions. Thank you!
Regards, Ivan
