Let $X$ be a Poisson random variable of rate $n$, then it is well-known that $\frac{X - n}{\sqrt{n}}$ converges weakly to the standard normal. It is probably not hard to show that in fact one gets a Berry-Esseen error bound of order $n^{-1/2}$, which implies that for $c$ real, $$\displaystyle |e^{-(n + cn^{1/4})}\sum_{l=0}^n \frac{(n + c n^{1/4})^l}{l!} - \Phi(0)| = O_c(n^{-1/4})$$. I would like to know if the above bound extends to the case when $c$ is a complex number. Also a reference for the Berry-Esseen theorem for Poisson random variable would be appreciated.
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