Exploring results of L´evy Continuity Theorem (relating moment generating functions and nth moments)

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I'm having trouble with the second part. My proof of the first part simply said that only Mx(t) = My(t) implies X and Y have the same CDF. E(X^n) = E(Y^n) does not always imply X and Y have the same CDF, as there are cases where the nth moments are equal (power series converge to same number) but the actual RVs X and Y have different distributions. I have no idea how to even approach the second part. It is very intimidating.

Answer 1

They give a hint!

Write $$E(X_a^n) = \int_0^\infty x^n\frac{1}{x\sqrt{2\pi}}e^{-\log(x)^2/2}(1+a\sin(2\pi\log(x)))dx$$

now transform to $s = \log(x)-n$ so that $ds = dx/x$ and $x = e^{n+s}$ and you get $$\begin{align}E(X_a^n) &= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{n(n+s)}e^{-(n+s)^2/2}(1+a\sin(2\pi(n+s)))ds \\ &= \frac{e^{n^2/2}}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-s^2/2}(1+a\sin(2\pi s))ds\end{align}$$ where in the second line we cleaned up what was in the exponential and used that $\sin(2\pi n + x) = \sin(x)$. Notice the term proportional to $s$ canceled out. Do you see why the final answer won't depend on $a$?

Also, I think they're looking for something different for part 1. Part 2 is showing that just because two distributions have all the same moments doesn't mean they're equal. All you do in your answer is state this. They want a reason that this doesn't violate the assumptions of the Levy continuity theorem. I don't know the statement of the theorem (never heard of a 'weak form' anyway), so I don't know for sure, but I expect the reason you might think it does superficially is that if the moments are all the same, then since the moment-generating function is a power series whose coefficients are the moments, then all the moments being the same should mean the moment-generating function is the same. Where's the hole in that argument?