Central Limit Theorem Proof: Expanding the Exponential

[PineApple2]

Hello. This is the most closely matching forum I found for this, so I hope my question fits here. I was looking at the following proof of the Central-Limit theorem:
http://physics.ucsc.edu/~peter/250/deriv_climit.pdf
and after Eq. (10) it says: "Expanding out the exponential in the last expression and comparing
powers of k one finds that the first few cumulants are..."
but I don't see how the equalities in the next lines stem from it.
Could someone please explicitly show that?

Thanks.

 

[chiro]

Hello. This is the most closely matching forum I found for this, so I hope my question fits here. I was looking at the following proof of the Central-Limit theorem:
http://physics.ucsc.edu/~peter/250/deriv_climit.pdf
and after Eq. (10) it says: "Expanding out the exponential in the last expression and comparing
powers of k one finds that the first few cumulants are..."
but I don't see how the equalities in the next lines stem from it.
Could someone please explicitly show that?

Thanks.

Hey PineApple2.

I think the best way for you would be to look at how the moments and cumulants are defined with respect to each other:

http://en.wikipedia.org/wiki/Cumulant
http://en.wikipedia.org/wiki/Moment_(mathematics)

Specifically with regard to your question:

http://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments