Dickman Approximation of Randomly Weighted Sums via Stein's Method

Abstract

We introduce a unified framework via Stein's method for bounding the Kolmogorov distance between the generalized Dickman distributions and the distribution of randomly weighted sums of non-negative integer-valued random variables that are conditionally independent given the weights. By utilizing size-bias couplings and a decomposition of the solution to the corresponding Stein equation into distinct bounded and non-increasing components, our approach yields non-asymptotic error bounds governed only by the quality of a discrete-to-continuous coupling. A similar bound is also obtained for the smooth Wasserstein-2 distance. We apply our abstract results to establish concrete, and in some cases optimal, rates of convergence across diverse probabilistic models. These applications range from settings with independent weights, such as randomly weighted sums of log-primes, to models with intricate dependency structures, including geometric sums over the spectra of the Circular Unitary Ensemble (CUE) and random weights generated by independent increments.