Dickman Approximation of weighted random sums in the Kolmogorov distance

TL;DR

This paper develops bounds in the Kolmogorov distance for approximating weighted sums of Bernoulli and Poisson variables by generalized Dickman distributions, using Stein's method, with applications in algorithms and random trees.

Contribution

It provides the first Kolmogorov distance bounds for these approximations and establishes their optimality, revealing phase transitions based on parameters.

Findings

Established new bounds in Kolmogorov distance for generalized Dickman approximation.

Proved the optimality of convergence rates with lower bounds.

Applied results to analyze Quickselect algorithm runtime and random tree depths.

Abstract

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 and the weighted depth in randomly grown simple increasing trees.