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- Title
CONVERGENCE IN A MULTIDIMENSIONAL RANDOMIZED KEYNESIAN BEAUTY CONTEST.
- Authors
GRINFELD, MICHAEL; VOLKOV, STANISLAV; WADE, ANDREW R.
- 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.
- Subjects
STOCHASTIC convergence; RANDOMIZATION (Statistics); BEAUTY contests; INDEPENDENCE (Mathematics); PROBABILITY theory
- Publication
Advances in Applied Probability, 2015, Vol 47, Issue 1, p56
- ISSN
0001-8678
- Publication type
Article
- DOI
10.1239/aap/1427814581