Covariance confidence bounds and parity certificates for stable count laws

Abstract

We construct an upper confidence bound for the operator norm of the normalized covariance deficit of a homogeneous real-stable count law on m categories with total count k, without knowing the mean or imposing a minimum category-probability assumption. This deficit is a rescaling of an established canonical dependence matrix for two balls drawn without replacement. Its positive semidefiniteness gives a diagonal domination property that controls rare categories and normalization error. Independent batches of full count observations suffice: the mean stage uses O(m(k-1)ε⁻² log(m/δ)/k) observations and the second stage uses O(m(k−1)²ε⁻² log(m/δ)). A stable degree-two subclass gives a matching linear category and inverse-square accuracy lower bound up to logarithms. Real-rooted one-dimensional projections turn a covariance bound and a reference Fourier margin into simultaneous positive-parity certificates. A classical sharp product inequality improves their quantitative floor and identifies feasible binary equality cases. We give a sharp positivity boundary on a binary subclass and stable laws with identical means and covariance but opposite parity. The constants in the confidence bound are conservative, and the observation model is classical labeled counts.