Open Access
Translator Disclaimer
July 2004 On the contraction method with degenerate limit equation
Ralph Neininger, Ludger Rüschendorf
Ann. Probab. 32(3B): 2838-2856 (July 2004). DOI: 10.1214/009117904000000171


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.


Download Citation

Ralph Neininger. Ludger Rüschendorf. "On the contraction method with degenerate limit equation." Ann. Probab. 32 (3B) 2838 - 2856, July 2004.


Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1060.60005
MathSciNet: MR2023025
Digital Object Identifier: 10.1214/009117904000000171

Primary: 60F05 , 68Q25
Secondary: 68P10

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

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3B • July 2004
Back to Top