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

I would like to show that, $\forall a \in \mathbb{R}_+^n$, $\parallel a\parallel_ 2 = 1 \Rightarrow \forall t\in \mathbb{R}_+^*$, $ p(\sum_{i=1}^n a_i X_i \leq t) \geq p(\sum_{i=1}^n \frac{1}{\sqrt{n}} X_i \leq t)$

If there wasn't the delta and $X_i$ was following a half gaussian distribution, the result would be a direct application of the theorem 2 from Yu 2011. However it is valid for a random variable with distribution, which is not my case here, and I struggle to see from their demonstration whether the presence of the delta is causing an issue. Is the result still true in my case ? If it is, how can I show it ?

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