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We consider distributional approximation by generalized Dickmandistributions, which appear in number theory, perpetuities, logarithmiccombinatorial structures and many other areas. We prove bounds in theKolmogorov distance for the approximation of certain weighted sums of Bernoulliand Poisson random variables by members of this family. While such results havepreviously been shown in Bhattacharjee and Goldstein (2019) for distances basedon smoother test functions and for a special case of the random variablesconsidered in this paper, results in the Kolmogorov distance are new. We alsoestablish optimality of our rates of convergence by deriving lower bounds. As aresult, some interesting phase transitions emerge depending on the choice ofthe underlying parameters. The proofs of our results mainly rely on the use ofStein's method. In particular, we study the solutions of the Stein equationcorresponding to the test functions associated to the Kolmogorov distance, andestablish their smoothness properties. As applications, we study the runtime ofthe Quickselect algorithm and the weighted depth in randomly grown simpleincreasing trees.

Dickman Approximation of weighted random sums in the Kolmogorov distance