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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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    Have you computed the characteristic function of $Z_n$ in this case? If so, you should show it in the question. If not, what is stopping you?2012-12-12
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    @did I have not and was looking for help in doing so2012-12-12
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    You do not say what is stopping you. So... let $N$ be Poisson with parameter $\lambda$, what is the characteristic function of $N$?2012-12-12
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    @did $e^{λ(e^{it}-1)}$, but I'm not sure how to use that to show convergence2012-12-12
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    Hold on, you will see... Now, what is the characteristic function of $(N-\lambda)/\sqrt{\lambda}$. (Please insert this in your question.)2012-12-12
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    @did I chose to use $Z_n=(N-\lambda)/\sqrt{\lambda}$ and $Φ_{Z_n}(j\omega)=E[e^{-j\omega Z_N}]$, but I am not certain how to proceed2012-12-12
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    Right, $Z_\lambda=(N_\lambda-\lambda)/\sqrt{\lambda}$ and $\mathbb E(\mathrm e^{\mathrm itN_\lambda})$ is what you wrote two comments ago, then what is $\mathbb E(\mathrm e^{\mathrm itZ_\lambda})$?2012-12-12
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    I used the probability generating function to obtain $\mathbb E(\mathrm e^{\mathrm itN_\lambda})$ but i'm not sure what the pmf is now for $Z_\lambda$2012-12-12
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    **PGF** for $Z_\lambda$.2012-12-12
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    @did I do not understand how to find the PGF for $Z_\lambda$2012-12-14
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    Forget the situation here and consider a random variable $Y$ and its PGF $\varphi_Y(t)=\mathbb E(\mathrm e^{\mathrm itY})$. Introduce $V=aY+b$. What is the PGF $\varphi_V(t)=\mathbb E(\mathrm e^{\mathrm itV})$? You must find a formula for $\varphi_V$ using $\varphi_Y$.2012-12-14
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    What now? $ $ $ $2012-12-15

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