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April 2003 Second phase changes in random $\boldsymbol{m}$-ary search trees and generalized quicksort: Convergence rates
Hsien-Kuei Hwang
Ann. Probab. 31(2): 609-629 (April 2003). DOI: 10.1214/aop/1048516530


We study the convergence rate to normal limit law for the space requirement of random $m$-ary search trees. While it is known that the random variable is asymptotically normally distributed for $3\le m\le 26$ and that the limit law does not exist for $m>26$, we show that the convergence rate is $O(n^{-1/2})$ for $3\le m\le 19$ and is $O(n^{-3(3/2-\alpha)})$, where $4/3<\alpha<3/2$ is a parameter depending on $m$ for $20\le m\le 26$. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the Berry--Esseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived.


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Hsien-Kuei Hwang. "Second phase changes in random $\boldsymbol{m}$-ary search trees and generalized quicksort: Convergence rates." Ann. Probab. 31 (2) 609 - 629, April 2003.


Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1021.60020
MathSciNet: MR1964943
Digital Object Identifier: 10.1214/aop/1048516530

Keywords: asymptotic normality , Convergeance rates , Local limit theorems , method of moments , phase change , QuickSort , search trees

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 2 • April 2003
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