Let ${\textstyle \{X_{1},\ldots ,X_{n}}\}$ be a sequence of independent and identical random variables. The distribution of $X_n$ is unknown.
Assuming that we know the distribution of the following summation:
$${S}\equiv \sum_{n=1}^{\infty}\frac{X_n}{n^2}$$
Would it be possible to find the distribution of $X_n$ from $S$ ?
Best Answer
Partial answer
The characteristic function of $X$ satisfies a functional equation, namely $$\prod_{n=1}^{+\infty} \phi_X(t/n^2) = \phi_S(t).$$ Solving this kind of equation is not simple.
If the distribution of $X$ is completely determined by its moments, or equivalently by its cumulants, then it is determined by the cumulants of $S$ since for every integer $d \ge 1$, $$\kappa_d(S) = \sum_{n=1}^{+\infty}\kappa_d(X/n^2) = \zeta(2d)\kappa_d(X).$$