Smoothness and Decay Properties of the Limiting Quicksort Density Function

Abstract

Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used by Quicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f , and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16.

Approximating the limiting Quicksort distribution

“…The proof is almost identical to the proof of the special case in [6], so we will omit some details. For any random variable Z, we abuse notation slightly and denote by SZ the random variable h Z,Z * (U ) = U Z + (1 − U )Z * + g(U ) where U , Z, and Z * are independent, with Z * L = Z and U ∼ unif(0, 1); thus SZ is a random variable with the distribution SL(Z).…”

“…However, there is no reason to believe that our method yields the best possible bounds, and the best constants for the special case in [6] may be smaller than the best constants in Theorem 2.1 here. ]…”

“…In [6] we gave bounds on the characteristic function of Y . The same method yields, more generally, bounds on the characteristic function of Z n for arbitrary Z 0 .…”

“…Rösler [15] showed that the moment generating function of the limiting distribution L(Y ) is everywhere finite. We have studied the limiting distribution further in [6], showing that L(Y ) has a density f which is infinitely differentiable, and that each derivative f (k) (y) is bounded and decays as y → ±∞ more rapidly than any power of |y| −1 . (This improves an earlier result by Tan and Hadjicostas [18].…”

“…Let X 2 := (E X 2 ) 1/2 denote the L 2 -norm, and let d 2 denote the metric on the space of probability distributions with finite variance defined by 5) taking the minimum over all pairs of random variables X and Y (defined on the same probability space) with L(X) = F and L(Y ) = G. Note that, using the coupling with X and Y independent, for any F and G each with zero mean and finite variance, 6) when L(X) = F and L(Y ) = G. Rösler [15] showed that if Z 0 has mean 0 and finite variance, then F n → F in the d 2 -distance with a geometric rate:…”