# Moment-Free Kunchenko Stochastic Polynomials via Empirical Characteristic Function

arXiv: https://arxiv.org/abs/2606.16289 · HTML: https://mlsanity.com/paper/2606.16289/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2026.

## Built on by (most cited first, 1 of 1)

- [Closed-form fractional radial links for elliptical Mahalanobis discriminant analysis](https://mlsanity.com/paper/2607.06089/) (2026)

## Abstract (arXiv)

We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.