# One sided point-wise Berry-Esseen like inequality for discrete variable

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...