Statistics of a multi-factor function from its Fourier transform

TL;DR

The paper establishes a method to derive the statistical moments of a multi-factor function on finite abelian groups directly from its Fourier transform, revealing new structural insights.

Contribution

It introduces the $m$-Coefficient/Index Annihilation Theorem, linking moments to Fourier coefficients with index sum constraints, enabling deeper analysis of multi-factor functions.

Findings

Derived moments from Fourier coefficients for functions on finite abelian groups.

Showed how distribution statistics like skew and kurtosis can be obtained from Fourier domain.

Presented multiple examples demonstrating the technique's applicability.

Abstract

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 $m$-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 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.