A functional limit theorem for the profile of search trees

Abstract

We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}{X}_{n,k}$ for $k=\lfloor \alpha \log n\rfloor$ in a certain range of $\alpha$. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.

Published in at http://dx.doi.org/10.1214/07-AAP457 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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