Moment statistics (reviewed)

Moment statistics refers to moments of random variables associated to a Sato-Tate group $G$, where the distribution of the random variable is defined using the pushforward of the Haar measure of $G$.

Standard random variables one can associated to $G$ include the coefficients $a_i$ of the characteristic polynomial $\sum a_ix^i$ of a random matrix $g\in G$, the $i$th symmetric function $e_i$ of the eigenvalues $\lambda_j$ of $g$, and the powers sums $s_i:=\sum \lambda_j^i$ which can be expressed as polynomials in the $a_i$ or $e_i$ via the Newton identities.

The moment sequence of $a_i$ (or any random variable) is the integer sequence $(\mathrm{E}[a_i^0], \mathrm{E}[a_i^1],\mathrm{E}[a_i^2]\ldots)$; the convention of including the zeroth moment $\mathrm{E}[a_i^0]=1$ facilitates the computation of moment sequences of products of independent random variables by taking binomial convolutions.