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With the central limit theorem, for an i.i.d. process $X_n$ (with mean $m$ and variance $σ^2$), the corresponding normalized sum process is: $ Z_n = \frac{S_n-nm}{σ\sqrt{n}} $ with $S_n = X_1+X_2+ . . . + X_n$. I know that this does indeed converge in distribution to a zero-mean unit-variance Gaussian. My question is if this is true for the Poisson process and why/why not? I am considering using the taylor expansion of the characteristic function to show whether or not it converges to that of a Gaussian, but I am not quite sure how.

Thanks!

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    What now? $ $ $ $2012-12-15

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