TITLE: Random variable $X^2$ determined by moments
QUESTION [5 upvotes]: Let $X$ be a real random variable, with standard normal distribution.
Is the distribution of $X^2$ determined by its moments?
In general, if $n \in \mathbb  N$, is the distribution of $X^n$ determined by its moments?

REPLY [2 votes]: Carleman's condition holds for $X$ hence the distribution of $X$ is determined by its moments.
Carleman's condition holds for $X^2$ hence the distribution of $X^2$ is determined by its moments. 
For every $n\geqslant1$, Krein's condition holds for $X^{2n+1}$ hence the distribution of $X^{2n+1}$ is not determined by its moments.
The distribution of $X^{4}$ is determined by its moments.
For every $n\geqslant3$, the distribution of $X^{2n}$ is not determined by its moments. 

Thus:

Let $n\geqslant1$. The distribution of $X^n$ is determined by its moments if and only if $n\in\{1,2,4\}$.