The Annals of Probability

On the contraction method with degenerate limit equation

Ralph Neininger and Ludger Rüschendorf

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Abstract

A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form $X\stackrel {\mathcal{L}}{=}X$. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2838-2856.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813632

Digital Object Identifier
doi:10.1214/009117904000000171

Mathematical Reviews number (MathSciNet)
MR2023025

Zentralblatt MATH identifier
1060.60005

Subjects
Primary: 60F05: Central limit and other weak theorems 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 68P10: Searching and sorting

Keywords
Contraction method analysis of algorithms recurrence recursive algorithms divide-and-conquer algorithm random recursive structures Zolotarev metric

Citation

Neininger, Ralph; Rüschendorf, Ludger. On the contraction method with degenerate limit equation. Ann. Probab. 32 (2004), no. 3B, 2838--2856. doi:10.1214/009117904000000171. https://projecteuclid.org/euclid.aop/1091813632


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