If I start with a Poisson distribution $X$ and I take $n$ random samples/observations from X, with mean of the samples $= \bar{x}$, the differences between each sample observation and the mean is $(x_1-\bar{x}),...,(x_n-\bar{x})$.
These deviations from the mean should have $\mu = 0$ (obviously) and $\sigma^2 = \sum_{i=1}^{n} \frac{(x_i - \bar{x})^2}{n}$
How can I show that these deviations from the mean of the Poisson are themselves a normal distribution? Do I just rely on central limit theorem with ufficiently large n?
Look at this histogram of the deviations from the mean, from 10000 samples from a Poisson with $\lambda=2$. Is it close to Normal?