The Annals of Probability

On the contraction method with degenerate limit equation

Ralph Neininger and Ludger Rüschendorf

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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

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

First available in Project Euclid: 6 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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