Abstract

In a continuous-time setting, Fill [2] 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. 1. The number of symbol comparisons used by QuickSort: Brief review of a limiting-distribution result In this section we briefly review the main theorem of [2]. An infinite sequence of independent and identically distributed keys is generated; each key is a random word (w1, w2,...) = w1w2 · · · , that is, an infinite sequence, or “string”, of symbols wi drawn from a totally ordered finite alphabet Σ. The common distribution µ of the keys (called a probabilistic source) is allowed to be any distribution over words, i.e., the distribution of any stochastic process with time parameter set {1, 2,...} and state space Σ. We know thanks to Kolmogorov’s consistency criterion (e.g.,

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