Using a recursive approach, we obtain a simple exact expression for the L^2-distance from the limit in Regniers (1989) classical limit theorem for the number of key comparisons required by QuickSort. A previous study by Fill and Janson (2002) using a
similar approach found that the d_2-distance is of order between n^{-1} log n and n^{-1/2}, and another by Neininger and Ruschendorf (2002) found that the Zolotarev zeta_3-distance is of exact order n^{-1} log n. Our expression reveals that the L^2-distance is asymptotically equivalent to (2 n^{-1} ln n)^{1/2}.
In a continuous-time setting, Fill (2010) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution,
but proved little about that limiting random variable Y -- not even that it is nondegenerate. We establish the nondegeneracy of Y. The proof is perhaps surprisingly difficult.
