Generalizes inference functions to distributional models using observation operators, establishes consistency and asymptotic normality, and derives a hierarchy of information bounds via the Hájek–Le Cam theorem.
Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem
5 Pith papers cite this work. Polarity classification is still indexing.
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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 a probability law is represented by a tempered distribution $T \in \mathcal{S}'$, on the same footing as a density, a distribution function, or a characteristic function. Information about the law is extracted by evaluating $T$ on test functions regularised by a given positive Schwartz kernel $\varphi \in \mathcal{S}$ -- the kernel serving as a probe, not as part of the law. 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.
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UNVERDICTED 5representative citing papers
Weak moment matching produces automatically Hampel-robust estimators with bounded influence functions inherited from kernel decay, demonstrated on Cauchy and t models with Monte Carlo comparisons.
The paper proves a weak transversality theorem showing that generic kernels in distributional models avoid high-codimension degeneracy strata encoding statistical failures.
Traces a four-stage conceptual chain from de Moivre's coefficient extraction to Schwartz distributions and proves a distributional version of the De Moivre-Laplace theorem in S'(R).
Statistical degeneracies in distributional models are geometric failures of transversality conditions on a kernel-induced feature map.
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