**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.

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- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2019-04-20 14:27:23

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**History:**(expand/hide all)

- 2021-01-01 17:59:45 by Andrew Sutherland
- 2021-01-01 12:37:51 by Andrew Sutherland
- 2019-04-20 14:27:23 by Kiran S. Kedlaya (Reviewed)
- 2016-05-04 03:04:44 by Andrew Sutherland

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