Advances in Applied Probability

Perpetuities in fair leader election algorithms

Ravi Kalpathy and Hosam Mahmoud

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Abstract

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 1 (2014), 203-216.

Dates
First available in Project Euclid: 1 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1396360110

Digital Object Identifier
doi:10.1239/aap/1396360110

Mathematical Reviews number (MathSciNet)
MR3189055

Zentralblatt MATH identifier
1291.60018

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Leader election recurrence functional equation fixed point contraction method metric space perpetuity weak convergence

Citation

Kalpathy, Ravi; Mahmoud, Hosam. Perpetuities in fair leader election algorithms. Adv. in Appl. Probab. 46 (2014), no. 1, 203--216. doi:10.1239/aap/1396360110. https://projecteuclid.org/euclid.aap/1396360110


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