Moment generating function of two variables

Question

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.

Answer 1

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?