Statistics of a multi-factor function from its Fourier transform

Battle Count: 55

The paper is a pure mathematical/statistical theory paper about Fourier analysis on finite abelian groups. It has minimal direct relevance to quantitative trading. However, the theoretical framework for understanding non-Gaussian distributions arising from sparse nonlinear interactions could conceptually inform risk modeling (fat tails, skewness). The feasibility constraint approach could theoretically be adapted for portfolio optimization with moment constraints. The coin-flip gambling example is tangentially related but is illustrative rather than applied. No financial data, trading strategies, or market models are discussed.

The theoretical framework is mathematically sophisticated, requiring knowledge of finite abelian groups, multidimensional DFT, Kronecker products, circulant matrices, and autoconvolution. Implementing the core Theorem 3.1 for small groups is straightforward (matrix powers of circulant matrices). However, scaling to large |G| or high moment orders m requires careful use of the autoconvolution recursion and sparsity exploitation. The index annihilation constraint requires group-theoretic operations (modulo addition in each dimension). The genetics and graph-theoretic applications add domain-specific complexity. No reference implementation is provided.

The paper provides complete mathematical derivations, explicit formulas for moments up to order 6, and detailed worked examples (1D DFT on Z_64, genetics on Z_2^13, coin-flip design on Z_2^4). All Fourier coefficients used in examples are listed in tables. The methodology is fully analytical and does not depend on proprietary data or code. However, no code repository is provided, and the genetics example references external data from prior publications [5, 30].