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Let us consider a distribution $\mathcal{L}$ on a finite set of integers (actually I'm even happy with any RV on the set $\{-1,0,1\}$) (with probability not to extremes : roughly around 1/4 for -1, 1 and 1/2 for 0).

I'm looking at the sum of such variables, so let set $Z_n=\sum_{i=1}^n X_i$ for a family of i.i.d variables $X$ drawn under $\mathcal{L}$ and I want to estimate the probability for $Z_n$ to be lower than 0. I would like a lower bound of the shape $ F_n(x) \geq \Phi(x)$ or some exponential lower bound (the upper bound is easy to get from sub-gaussian variables or Hoeffding bounds).

Hence, I tried to approximate the cumulative distribution $F_n(x)$ of the (conveniently normalised, recentered) $Z_n$ by the normal CDF $\Phi(x)$ and evaluate this on $x=\frac{\sqrt{n}E(X)}{\sigma}$, where $\sigma$ is the standard deviation of the $X_i$.

My problem is that I don't want an actual Berry-Esseen like inequality, but a one-sided one (plus the uniform decay in $K/\sqrt{n}$ is too weak for my purpose since I ultimately want to look at $F_n(x)^\alpha$ for very large $\alpha$). Going further than Berry-Esseen I looked at Edgeworth expansions, but since I'm on the tails of the distributions, I can't really figure out what are the signs of the remainders in the expression...

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