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Dickman approximation of weighted sums of independent random variables in the Kolmogorov distance
Publikationstyp
Journal Article
Date Issued
2025-10-20
Sprache
English
Volume
35
Issue
5
Start Page
3271
End Page
3309
Citation
Annals of Applied Probability 35 (5): 3271-3309 (2025)
Publisher DOI
Scopus ID
Publisher
Institute of Mathematical Statistics
We 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.
Subjects
60F05
60G50
Dickman distribution
Kolmogorov distance
Stein’s method
weighted Bernoulli sums
DDC Class
510: Mathematics