Abstract
We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U [0,1]d random particle. We show that the limiting configuration contains N - 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.
Original language | English |
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Pages (from-to) | 57-82 |
Number of pages | 26 |
Journal | Advances in Applied Probability |
Volume | 47 |
Issue number | 1 |
DOIs | |
Publication status | Published - 31 Mar 2015 |
Keywords
- Keynesian beauty contest
- radius of gyration
- rank-driven process
- sum of squared distances