Electronic Journal of Probability

On the Moment-Transfer Approach for Random Variables Satisfying a One-Sided Distributional Recurrence

Che-Hao Chen and Michael Fuchs

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The moment-transfer approach is a standard tool for deriving limit laws of sequences of random variables satisfying a distributional recurrence. However, so far the approach could not be applied to certain "one-sided" recurrences with slowly varying moments and normal limit law. In this paper, we propose a modified version of the moment-transfer approach which can be applied to such recurrences. Moreover, we demonstrate the usefulness of our approach by re-deriving several recent results in an almost automatic fashion.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 30, 903-928.

Accepted: 10 May 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C05: Trees
Secondary: 60F05: Central limit and other weak theorems 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

distributional recurrence moment-transfer approach central limit theorem analysis of algorithms

This work is licensed under aCreative Commons Attribution 3.0 License.


Chen, Che-Hao; Fuchs, Michael. On the Moment-Transfer Approach for Random Variables Satisfying a One-Sided Distributional Recurrence. Electron. J. Probab. 16 (2011), paper no. 30, 903--928. doi:10.1214/EJP.v16-885. https://projecteuclid.org/euclid.ejp/1464820199

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  • A. Bagchi and A. K. Pal. Asymptotic normality in the generalized Polya-Eggenberger urn model, with an application to computer data structures. SIAM Journal on Algebraic and Discrete Methods 6 (1985), 394–405.
  • Z.-D. Bai, H.-K. Hwang, W.-Q. Liang. Normal approximations of the number of records in geometrically distributed random variables. Random Structures and Algorithms 13 (1998), 319–334.
  • H.-H. Chern, M. Fuchs, H.-K. Hwang. Phase changes in random point quadtrees. ACM Transactions on Algorithms 3 (2007), 51 pages.
  • H.-H. Chern and H.-K. Hwang. Phase changes in random m-ary search trees and generalized quicksort. Random Structures and Algorithms 19 (2001), 316–358.
  • H.-H. Chern, H.-K. Hwang, T.-H. Tsai. An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms. Journal of Algorithms 44 (2002), 177–225.
  • L. Devroye. Universal limit laws for depths in random trees. SIAM Journal on Computing 28 (1999), 409–432.
  • P. D. T. A. Elliott. Probabilistic Number Theory I. Central Limit Theorems. Springer, New-York (1979).
  • P. Flajolet and T. Lafforgue. Search costs in quadtrees and singularity perturbation asymptotics. Discrete and Computational Geometry 12 (1994), 151–175.
  • P. Flajolet and R. Sedgewick. An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading, MA (1996).
  • P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press (2009).
  • V. Goncharov. On the field of combinatory analysis. Soviet Math. IZv., Ser.Math. 8 (1944), 3–48.
  • H.-K. Hwang. Phase changes in random recursive structures and algorithms (a brief survey). "Proceedings of the Workshop on Probability with Applications to Finance and Insurance", World Scientific 8 (2004), 82–97.
  • H.-K. Hwang and R. Neininger. Phase change of limit laws in the quicksort recurrences under varying toll functions. SIAM Journal on Computing 31 (2002), 1687–1722.
  • A. Iksanov, A. Marynych, M. Moehle. On the number of collisions in beta(2,b)-coalesents. Bernoulli 15 (2009), 829–845.
  • M. Kuba and A. Panholzer. Analysis of insertion costs in priority trees. "Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithmics and Combinatorics", SIAM Philadelphia (2007), 175–182.
  • H. M. Mahmoud. Evolution of Random Search Trees. Wiley, New York (1992).
  • H. M. Mahmoud and B. Pittel. On the joint distribution of the insertion path length and the number of comparisons in search trees. Discrete Applied Mathematics 20 (1988), 243–251.
  • R. Neininger and L. Rueschendorf. On the contraction method with degenerate limit equation. The Annals of Probability 32 (2004), 2838–2856.
  • A. Panholzer. Analysis for some parameters for random nodes in priority trees. Discrete Mathematics and Theoretical Computer Science 10 (2008), 1–38.
  • A. Panholzer and H. Prodinger. Average case analysis priority trees: a structure for priority queue administration. Algorithmica 22 (1998), 600–630.