Moment statistics (reviewed)

Moment statistics refers to moments of random variables associated to a Sato-Tate group $G$.

The moment $\mathrm{M}_n[a_i]:=\mathrm{E}[a_i^n]$ denotes the $n$th moment of the coefficient $a_i$ in the characteristic polynomial $\sum a_ix_i$ of a random matrix $g\in G$ under the Haar measure of a topological group on $G$; equivalently, the $n$th moment of the $i$th symmetric function of the eigenvalues $\lambda_j$ of $g$.

Similarly, the moment $\mathrm{M}_n[s_i]$ denotes the $n$th moment of the $i$th power sum $s_i:=\sum \lambda_j^i$. The $a_i$ and $s_i$ are related via the Newton identities. Using these one can derive the moment sequences of the $a_i$ from those of the $s_i$, and conversely.