Let $X_1,X_2,\ldots,X_n$ be i.i.d ~$ Unif (-\frac{1}{2},\frac{1}{2})$ and $S_n= \sum_{i=1}^n X_i$. Let $a \in \mathbb{R}: \frac{1}{n} \leq a \leq \frac{2}{n}$
My aim is to:
1) Find the asymptotic behavior of the following cumulative distributive function: $$ p_n= P\bigg(\bigg|\frac{S_n}{\sqrt n}\bigg| \leq \frac{1}{a\sqrt n}- \frac{\sqrt n}{2} \bigg) $$ 2) Find $a$ that maximizes $p_n$.
Through Central Limit theorem and using Berry-Essen theorem, I'm only able to find the upper bound for $p_n$.
Can we characterize the lower bound as well ? I believe $p_n$ behaves as $\Theta(1/\sqrt n)$