Density functions for QuickQuant and QuickVal

TL;DR

This paper establishes the existence, boundedness, positivity, and tail decay properties of the limiting density functions for the number of key comparisons in QuickQuant, providing insights into its probabilistic behavior.

Contribution

It proves the Lipschitz continuity, positivity, and superexponential tail decay of the limiting densities for QuickQuant's key comparison counts, with precise tail asymptotics.

Findings

Density functions are Lipschitz continuous and bounded by 10.

Density functions are positive for x > min{t, 1 - t}.

Right tail exhibits superexponential decay with specific asymptotics.

Abstract

We prove that, for every $0 \leq t \leq 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the $t$th quantile in a randomly ordered list has a Lipschitz continuous density function $f_t$ that is bounded above by $10$. Furthermore, this density $f_t(x)$ is positive for every $x > \min\{t, 1 - t\}$ and, uniformly in $t$, enjoys superexponential decay in the right tail. We also prove that the survival function $1 - F_t(x) = \int_x^{\infty}\!f_t(y)\,\mathrm{d}y$ and the density function $f_t(x)$ both have the right tail asymptotics $\exp [-x \ln x - x \ln \ln x + O(x)]$. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.