I want to lower bound
\begin{align} \mathbb{P} \left( -\epsilon \leq \sum_{i=1}^n X_i \leq 0 \right) &= \mathbb{P} \left( - \frac{\epsilon}{\sqrt{\sum_{i=1}^n \sigma_i^2}} \leq \frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n \sigma_i^2} \leq 0 \right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1) \end{align} where $\{X_i\}$ are mean zero, independent random variables with $\mathbb{E}[X_i^2] = \sigma_i^2 \leq H^2$ and $\mathbb{E}[|X_i|^3] = \lambda_i^3 \leq H^3$.
From Equation $3$ in Berry Esseen, I have that
\begin{align} \sup_{x \in \mathbb{R}} |F_n(x) - \Phi(x)| \leq C_0 \left( \sum_{i=1}^n \sigma_i^2 \right)^{-3/2} \sum_{i=1}^n \lambda_i^3 \end{align} Loosening the bound above gives \begin{align} \sup_{x \in \mathbb{R}} |F_n(x) - \Phi(x)| \leq \frac{C_0}{\sqrt{n}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2) \end{align} Now I can write $(1)$ as
\begin{align} F_n(0) - F_n \left(- \frac{\epsilon}{\sqrt{\sum_{i=1}^n \sigma_i^2}} \right) \end{align} which by $(2)$ can be lower bounded as
\begin{align} \Phi(0) - \frac{2C_0}{\sqrt{n}} - \Phi\left ( - \frac{\epsilon}{\sqrt{\sum_{i=1}^n \sigma_i^2}} \right ) &\geq \frac{1}{2} - \frac{2C_0}{\sqrt{n}} - \Phi\left ( - \frac{\epsilon}{H \sqrt{n}} \right ) \end{align} Can the last expression be improved to something like $K/\sqrt{n}$ for some constant $K$ for large $n$? I feel like I need some approximations to normal CDF $\Phi(x)$ for small $x$.