On the tails of the limiting QuickSort density

TL;DR

This paper establishes nearly matching asymptotic bounds for both tails of the limiting QuickSort density and its derivatives, improving previous results on the distribution function.

Contribution

It provides new asymptotic bounds for the tails and derivatives of the QuickSort density, strengthening earlier findings on the distribution function.

Findings

Nearly matching bounds for left and right tails of the density

Bounds on derivatives of the density of all orders

Enhanced understanding of the QuickSort limiting distribution

Abstract

We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density f that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function F. Furthermore, we obtain similar bounds on absolute values of derivatives of f of each order.