Product of CDFs at different points Suppose that $X$ is a continuous random variable with finite moments, and $\mathbb a=(a_i) \in \mathbb R^n$. In my calculations, I have reached an expression as
$$ f(\mathbb a )=\prod_i \Pr \{ X\le a_i\} $$
but I am not sure what to do with it to make it a simple expression (i.e. a CDF of a univariate random variable or a limiting theorem that make the whole thing tractable at least asymptotically, etc.). Let's suppose that $n$ is large enough.
One idea was to take another random variable $ Y_i = X - a_i$ , then $$ f(\mathbb a )= \prod_i \Pr \{ Y_i\le 0\} = \Pr \{ \max_i (Y_i)\le 0\} $$ and try to use results from Extreme Value Theory but I only know this theory for i.i.d. case (here we have different first moments).

I can only help you with a special case: Let's X is Gumbel-distributed
$$\mathrm{Pr}\{X\leq a_{i}\}=e^{-e^{-\frac{x-a_{i}}{\beta}}}$$
with the same variance $\beta$. Then
$$f(a)=\prod_{i=1}^{n}\mathrm{Pr}\{X\leq a_{i}\}=\prod_{i=1}^{n}e^{-e^{-\frac{x-a_{i}}{\beta}}}=e^{-e^{-\frac{x}{\beta}}\left(\sum_{i=1}^{n}e^{\frac{a_{i}}{\beta}}\right)}=e^{-e^{-\frac{x-\beta\log\left(\sum_{i=1}^{n}e^{\frac{a_{i}}{\beta}}\right)}{\beta}}}=\mathrm{Pr}\left\{ X\leq \bar{a}\right\} $$
where the new mean
$$\bar{a}=\beta\log\left(\sum_{i=1}^{n}e^{\frac{a_{i}}{\beta}}\right)$$
is often called the mellowmax.
