Moment generating function of two variables I am able to do all the parts except the very last. I have been trying to coax the differential equation $$\frac{M'}{M}=t$$ or something to that effect but I don't see how I can achieve this. Hints would be much appreciated.

We have $$M(2t)=M(t)^3 M(-t)$$ and $$M(t)=M(-t)$$, so $$M(2t)=M(t)^4$$. Hence we have a functional equation for $$m(t)=t^{-2}\log M(t)$$ (let $$m(0)=\frac12s^2=\frac12$$): $$m(2t)=m(t)\quad\text{for all }t\tag{1}$$ But we know $$m(t)=\frac12+o(1)\text{ for small }t\tag{2}$$ from the expansion of $$M(t)$$.
Now equations (1) and (2) together implies $$m$$ is constant $$\frac12$$. Can you see why?
• So letting $o(1) = f_1(t)$ we can see that $f_1(t)=f_1(2t) \implies f$ constant and $m(0) = 1/2$ gives the result? Question: how did we get equation (2) – user3184807 May 22 at 10:13
• Not quite. By (1), $m(t)=m(t/2)=m(t/4)=\dots$. By (2), we have $m(t/2^n)\to\frac12$. The only way to reconcile these two is if $m(t)=\frac12$, any other value would violate the limit. – user10354138 May 22 at 10:16