Bhattacharjee, ChinmoyChinmoyBhattacharjeeSchulte, MatthiasMatthiasSchulte2025-11-032025-11-032025-10-20Annals of Applied Probability 35 (5): 3271-3309 (2025)https://hdl.handle.net/11420/58427We consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the approximation of certain weighted sums of Bernoulli and Poisson random variables by members of this family.While such results have previously been shown in Bhattacharjee and Goldstein (2019) for distances based on smoother test functions and for a special case of the random variables considered in this paper, results in the Kolmogorov distance are new. We also establish optimality of our rates of convergence by deriving lower bounds. As a result, some interesting phase transitions emerge depending on the choice of the underlying parameters. The proofs of our results mainly rely on the use of Stein’s method. In particular, we study the solutions of the Stein equation corresponding to the test functions associated to the Kolmogorov distance, and establish their smoothness properties. As applications, we study the runtime of the Quickselect algorithm, an edge-length statistic of a long-range percolation model, and the weighted depth in randomly grown simple increasing trees.en2168-87372025532713309Institute of Mathematical Statistics60F0560G50Dickman distributionKolmogorov distanceStein’s methodweighted Bernoulli sumsNatural Sciences and Mathematics::510: MathematicsJournal Article10.1214/25-AAP2194Journal Article