Sum of i.i.d discrete random variables, anything better than CLT?

by Changyu Dong   Last Updated August 14, 2019 09:20 AM - source

I know by CLT the sum of i.i.d random variables converges to the normal distribution, but is there any way to get a more refined distribution knowing the distribution of the individual random variable? The problem I have is the following:

We have a randomized function $f: \mathbb{Z}^*\rightarrow \mathbb{Z}^*$, and we know the Probability mass function of its output $f(n)$ is: $$ \left\{ \begin{array}{ll} Pr[f(n)=0]=(1-\frac{1}{2})^n &\\ Pr[f(n)=i]=(1-\frac{1}{2^{i+1}})^n \prod_{j=0}^{i-1}(1-(1-\frac{1}{2^{j+1}})^n), &1\le i\le w-1\end{array} \right. $$

Then let $m\in \mathbb{N}$, let $z_1,...z_m$ be independent samples from $f(n)$, what is the PMF of $\mathcal{Z}=\sum_{i=1}^m z_i$?

The problem why I cannot use CLT is that our $m$ is limited ($10^3 - 10^4$), and there is a question how close the distribution of $\mathcal{Z}$ to the normal distribution. Currently, the only bound I know is the Berry–Esseen theorem, in which the approximation is measured by the Kolmogorov–Smirnov distance, and is $O(\frac{1}{\sqrt{m}})$. This is too big for me. However, the bound in Berry–Esseen theorem is the worst-case bound (we don't care too much about worst-case TBH) and is loose, so maybe we can get a more precise result.

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