Statistics of a multi-factor function from its Fourier transform

For a phenomenon $\pmb{f}$ that is a function of $\mathit{n}$ factors, defined on a finite abelian group $\mathcal{G}$, we derive its population statistics solely from its Fourier transform $\hat{\pmb{f}}$. Our main result is an $\mathit{m-Coefficient/Index Annihilation Theorem}$: the $\mathit{m}$th moment of $\pmb{f}$ becomes a series of terms, each with precisely $\mathit{m}$ Fourier coefficients -- and surprisingly, the coefficient $\mathit{indices}$ in each term sum to zero under group addition. This condition acts like a filter, limiting which terms appear in the Fourier domain, and can reveal deeper relationships between the variables driving $\pmb{f}$. These techniques can also be used as an analytical/design tool, or as a feasibility constraint in search algorithms. For functions defined on $\mathbb{Z}_2^n$, we show how the skew, kurtosis, etc. of a binomial distribution can be derived from the Fourier domain. Several other examples are presented.