I have a question related to this one. $X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a folded Gaussian and a delta in $0$, both with weight $1/2$.

I have a vector $a$ with $\parallel a \parallel_ 2 = 1$, and I want to find a function $F : \mathbb{R} \rightarrow [0,1]$ so that $\forall a$ with $\parallel a \parallel_ 2 = 1$, $\forall t \in \mathbb{R}$, $p(\sum_{i=1}^n a_i X_i \leq t) \geq F(t)$, i.e $F$ is the CDF of an upper bound of $\sum_{i=1}^n a_i X_i$ for the stochastic order.

For example, with $Y_i$ iid random variable distributed as a folded Gaussian, using the fact that $p(X_i\leq t)\geq p(Y_i\leq t)$ and the theorem 2 from Yu (2011), we get by conditioning on the numbers of $0$ in $(X_i)$ that

$p(\sum_{i=1}^n a_i X_i \leq t) \geq \left( \frac{1}{2} \right)^n + \sum_{k=1}^n \left( \frac{1}{2} \right)^n C_n^k p(\sum_{i=1}^k \frac{1}{\sqrt{k}} Y_i \leq t)$

Can we find a tighter bound, especially in the tail of the distribution of $\sum_{i=1}^n a_i X_i$ ?

Yu, Y. (2011). Some stochastic inequalities for weighted sums. Bernoulli, 17(3). https://arxiv.org/abs/0910.0544

4more comments