Statistics of a multi-factor function from its Fourier transform
Provides a new theoretical tool for analyzing multi-factor functions in fields like combinatorics and search algorithms, but the results are domain-specific and incremental.
The paper derives population statistics of a multi-factor function from its Fourier transform, introducing an m-Coefficient/Index Annihilation Theorem that filters Fourier terms based on index sums. This enables analytical derivation of moments like skew and kurtosis for functions on finite abelian groups, demonstrated on binomial distributions.
For a phenomenon~$\boldsymbol{f}$ that is a function of~$n$ factors, defined on a finite abelian group $G$, we derive its population statistics solely from its Fourier transform~$\hat{\boldsymbol{f}}$. Our main result is an \emph{$m$-Coefficient/Index Annihilation Theorem}: the $m$th moment of~$\boldsymbol{f}$ becomes a series of terms, each with precisely $m$ Fourier coefficients -- and surprisingly, the coefficient \emph{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~$\boldsymbol{f}$. 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.