Density functions for QuickQuant and QuickVal

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 ft that is bounded above by 10. Furthermore, this density ft(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 − Ft(x) = ∫∞ x ft(y) dy and the density function ft(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.

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