Density functions for QuickQuant and QuickVal

09/29/2021

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by   James Allen Fill, et al.

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We prove that, for every 0 ≤ t ≤ 1, 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 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) = ∫_x^∞f_t(y) dy and the density function f_t(x) both have the right tail asymptotics exp [-x ln x - x lnln x + O(x)]. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.