Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem

TL;DR

This paper introduces a generalized probabilistic framework using weak moments, cumulants, and a weak CLT to analyze distributions outside classical moment-based methods, with applications to distributions like Student's t and Cauchy.

Contribution

It develops a systematic algebra of weak cumulants, solves the weak moment problem, and formulates a weak central limit theorem applicable where classical methods fail.

Findings

Established algebra of weak cumulants

Proved weak moment problem with various kernel conditions

Formulated weak CLT with convergence to Gaussian

Abstract

Many important statistical models fall outside classical moment-based methods due to the non-existence of moments or moment generating functions. We propose a generalised probabilistic framework in which densities are replaced by pairs $(T,\varphi)$, where $T \in \mathcal{S}'(\mathbb{R})$ is a tempered distribution and $\varphi \in \mathcal{S}(\mathbb{R})$ is a Schwartz kernel. Expectations are defined via the action of distributions on regularised test functions, yielding well-defined weak moments, weak characteristic functions, and weak cumulants of all orders. These extend classical quantities and retain key algebraic properties such as additivity under independence and natural affine transformation rules. The main results are: (i) a systematic algebra of weak cumulants; (ii) a weak moment problem where existence of all moments holds unconditionally and uniqueness depends on the kernel, with uniqueness results under Gaussian kernels (via Hermite completeness), positive Schwartz kernels with an exponential tail bound and square-integrable densities (via a Carleman-type criterion), and kernels with exponential decay (via Denjoy-Carleman quasi-analyticity); and (iii) a weak central limit theorem formulated as convergence of weak characteristic functions to a Gaussian limit, covering cases where the classical theorem fails. The framework is illustrated with Student's $t$, stable, and hyperbolic distributions. As a statistical consequence, the weak first moment yields a consistent estimator of the location parameter in the Cauchy model, where no classical moment-based estimator exists. A full statistical treatment is given in a companion paper.