Paper: arXiv 2605.02248

Authors: Matthew A. Herman, Stephen Doro

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 \textit{$m$-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 \textit{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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on group theory, multidimensional Fourier transforms, Kronecker products, and circulant matrix algebra to derive formal statistical identities. While it provides a rigorous theoretical framework for analyzing the moments of functions, the empirical component is conceptual and pedagogical rather than a demonstration on market datasets or backtested strategies.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive statistics from Fourier transform] --> B[Input: Function f on finite abelian group G]
    B --> C[Method: M-Coefficient/Index Annihilation Theorem]
    C --> D[Process: Zero-sum index constraint on Fourier coefficients]
    D --> E[Application: Moment calculation & feasibility constraints]
    E --> F[Outcome: Statistical moments & binomial distribution analysis]

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0 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.