Open Access
2023 Density functions for QuickQuant and QuickVal
James Allen Fill, Wei-Chun Hung
Author Affiliations +
Electron. J. Probab. 28: 1-50 (2023). DOI: 10.1214/22-EJP899


We prove that, for every 0t1, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the tth quantile in a randomly ordered list has a Lipschitz continuous density function ft that is bounded above by 10. Furthermore, this density ft(x) is positive for every x>min{t,1t} and, uniformly in t, enjoys superexponential decay in the right tail. We also prove that the survival function 1Ft(x)=xft(y)dy and the density function ft(x) both have the right tail asymptotics exp[xlnxxlnlnx+O(x)]. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.

Funding Statement

Research of both authors supported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics.


We thank Svante Janson, the Editor of Electronic Journal of Probability, two Associate Editors, and a referee for helpful comments on earlier versions of this paper; they led to significant improvements.


Download Citation

James Allen Fill. Wei-Chun Hung. "Density functions for QuickQuant and QuickVal." Electron. J. Probab. 28 1 - 50, 2023.


Received: 30 September 2021; Accepted: 26 December 2022; Published: 2023
First available in Project Euclid: 5 January 2023

Digital Object Identifier: 10.1214/22-EJP899

Primary: 68P10
Secondary: 60C05 , 60E05

Keywords: asymptotic bounds , convolutions of distributions , densities , integral equations , large deviations , Lipschitz continuity , moment generating functions , perfect simulation , QuickQuant , QuickSelect , QuickVal , searching , tails of densities , tails of distributions

Vol.28 • 2023
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