Biasing with an independent increment: Gaussian approximations and proximity of Poisson mixtures

By exploiting the well-known observation that size-biasing or zero-biasing an infinitely divisible random variable may be achieved by adding an independent increment, combined with tools from Stein’s method for compound Poisson and Gaussian approximations, we establish three sets of approximation results: (a) bounds on the proximity of Poisson mixtures with infinitely divisible mixing distributions, (b) central limit theorems with explicit error bounds for sums of associated or negatively associated random variables which do not require boundedness of the underlying distributions, and (c) a Gaussian approximation theorem under a vanishing third moment condition. These exploit biasing by an independent increment directly, via an intermediate compound Poisson approximation, and through a convex ordering argument, respectively. Applications include a Dickman-type limit theorem, simple random sampling and urn models with overflow.

💡 Research Summary

This paper presents a sophisticated mathematical framework for approximating complex probability distributions by leveraging the property that biasing operations—specifically size-biasing and zero-biasing—for infinitely divisible random variables can be reformulated as the addition of an independent increment. By utilizing this structural simplification, the authors integrate Stein’s method for both compound Poisson and Gaussian approximations to establish three significant theoretical results.

The first contribution involves establishing precise bounds on the proximity of Poisson mixtures where the mixing distributions are infinitely divisible. This provides a quantitative measure of how closely these complex mixtures can be approximated by simpler Poisson structures, which is essential for controlling errors when approximating complex mixture models.

The second major achievement is the development of Central Limit Theorems (CLTs) with explicit error bounds for sums of associated or negatively associated random variables. A critical advancement here is the removal of the requirement for the underlying distributions to be bounded. This allows for the application of these theorems to a much broader class of distributions, including those with heavier tails, which is essential for realistic statistical modeling and inference where boundedness cannot be assumed.

The third contribution is a Gaussian approximation theorem that holds under a vanishing third moment condition. To prove this, the authors employ a convex ordering argument, demonstrating that the complexity of the distribution can be effectively managed through the lens of convex orderings to achieve Gaussian convergence.

The methodological strength of this paper lies in its multi-faceted approach: using direct independent increment biasing, intermediate compound Poisson approximations, and convex ordering arguments. These techniques converge to provide a robust toolkit for analyzing convergence in probability theory. The practical implications of this work are evident in its applications to the Dickman-type limit theorem, as well as in the analysis of simple random sampling and urn models involving overflow. Ultimately, the paper provides a powerful new lens through which the convergence and approximation of complex, dependent random variables can be rigorously studied.