Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$. what are some general techniques for showing that $Y$ has a normal distribution? That is,
$$P(X\leqslant a) = \frac{1}{\sqrt{2\pi}}\int\_{-\infty}^a e^{-t^2/2}\,dt.$$


The standard technique I know is to compute the moments or [cumulants](https://en.wikipedia.org/wiki/Cumulant) and then use the fact that the normal distribution is characterized by its moments/cumulants. Are there any other general techniques, and what are their advantages and disadvantages?




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The motivation for this question is number theoretic (as with [this related question](https://mathoverflow.net/questions/102964/convergence-of-moments-implies-convergence-to-normal-distribution)), hence the number theory tags. Specifically, the motivating theorem is Selberg's central limit theorem (first published in [Tsang's thesis](https://mathscinet.ams.org/mathscinet-getitem?mr=2633927), see also [this article](https://mathscinet.ams.org/mathscinet-getitem?mr=3832861) of Radziwill-Soundararajan) which states that for large $T$, the real valued random variable on $[T,2T]$ given by $t \mapsto \log|\zeta(\tfrac12 + it)|$ is approximately normally distributed with mean $0$ and variance $\frac{1}{2}{\log\log T}$.


Both the proofs I know of (Selberg's original proof, and that of Radziwill-Soundararajan) use the method of moments. Morally speaking, the analytic-number-theoretic input goes into showing that the contributions from the zeros of zeta can be controlled, and hence at least for the distributional question with $t \in [T,2T]$,
$$\log|\zeta(\tfrac12 + it)| \simeq \Re\sum\_{p\leqslant T^{o(1)}} \frac{1}{p^{1/2 + it}}.$$
One can then compute the moments of the right hand side and show that as $T \to \infty$, the moments appropriately normalized converge to the moments of a standard Gaussian.


The hope is to see if there's a way to prove Selberg's CLT in a situation where the moments are harder to compute, and so the method of moments may not be tractable.



Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
There are multiple books about ways to characterize the normal distribution. For instance, Bryc’s [book](https://homepages.uc.edu/%7Ebrycwz/probab/charakt/charakt.pdf) starts with Herschel-Maxwell’s theorem:



> 
> If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.
> 
> 
> 


He immediately notes that one can strengthen this to Polya’s theorem:



> 
> If $X$ and $Y$ are independent variables, and rotations of $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$
> and $Y$ are both normal.
> 
> 
> 


Perhaps somewhere in such books you’ll find a characterization that avoids moments but is number-theoretically tractable.


