We have $n$ standard Gaussian random variables $x_i \sim N(0,1)$ and constants $a_i \in \mathbb{Z}_{\geq 0}$. We select the $k$ smallest $x_i$ in magnitude in a set $K$ and define $Z:= \sum_{i \in K} a_ix_i$.
What is the probability that $Z\geq 0$: $Pr(Z \geq 0)$.
I was wondering if this is even computable. Since we multiply the $x_i$ with a constant we obtain $a_ix_i \sim N(0, a_i^2)$. But now we cannot simply derive the probability distribution of the sum of $a_ix_i$, since we do not choose them randomly but select the smallest $x_i$ in magnitude.
Hence, I was thinking about order statistics. But for order statistics we need that the random variables are iid, which they are not since they have a different variance.
Am I correct? Or does there exist a way.
I would even be happy with something like $Pr(Z\geq 0) \geq ...$, but unfortunately since we do not know anything about $a_i$ I don't see how something can be derived.