Statistics of a multi-factor function from its Fourier transform
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916016320151552 |
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| author | Herman, Matthew A. Doro, Stephen |
| author_facet | Herman, Matthew A. Doro, Stephen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02248 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Statistics of a multi-factor function from its Fourier transform Herman, Matthew A. Doro, Stephen Statistics Theory Discrete Mathematics Signal Processing Genomics Statistical Finance 62H10, 42B05 C.4; G.1.2; G.1.3; G.2; G.3; I.2.8; J.2; J.3; J.4 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. |
| title | Statistics of a multi-factor function from its Fourier transform |
| topic | Statistics Theory Discrete Mathematics Signal Processing Genomics Statistical Finance 62H10, 42B05 C.4; G.1.2; G.1.3; G.2; G.3; I.2.8; J.2; J.3; J.4 |
| url | https://arxiv.org/abs/2605.02248 |