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Limit theorem for the Robin Hood game.

Omer Angel1, Anastasios Matzavinos2, Alexander Roitershtein3

  • 1Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada.

Statistics & Probability Letters
|July 10, 2019
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Summary
This summary is machine-generated.

This study analyzes the Robin Hood game, a probability problem involving removing balls from an urn. The research shows that the time to remove all balls follows a Fréchet distribution under random selection.

Keywords:
Fréchet’s distributioncoupon collector problemextreme order statisticslimit theoremsurn scheme

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Area of Science:

  • Probability theory
  • Stochastic processes
  • Combinatorics

Background:

  • The Robin Hood game models a scenario where items are removed from a collection over time.
  • Understanding the time dynamics of such processes is crucial in various fields.

Purpose of the Study:

  • To analyze the asymptotic behavior of the time required to empty an urn in the Robin Hood game.
  • To determine the limiting distribution of this random time under uniform random selection.

Main Methods:

  • Mathematical modeling of the Robin Hood game using an urn scheme.
  • Application of probability theory to derive limit theorems.
  • Analysis of the random variable representing the total time to remove all balls.

Main Results:

  • A limit theorem is established for the random time T, representing the total time to remove all balls.
  • It is shown that T multiplied by n (where n is the number of days) converges in distribution to a Fréchet distribution.
  • This convergence holds when balls are selected uniformly at random.

Conclusions:

  • The study provides a precise mathematical description of the long-term behavior of the Robin Hood game.
  • The Fréchet distribution characterizes the time required to clear the urn under random sampling.
  • This finding has implications for understanding random removal processes.