### Convergence in a multidimensional randomized Keynesian beauty contest

#### 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.

#### Article information

Source
Adv. in Appl. Probab., Volume 47, Number 1 (2015), 57-82.

Dates
First available in Project Euclid: 31 March 2015

https://projecteuclid.org/euclid.aap/1427814581

Digital Object Identifier
doi:10.1239/aap/1427814581

Mathematical Reviews number (MathSciNet)
MR3327315

Zentralblatt MATH identifier
1318.60100

#### Citation

Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R. Convergence in a multidimensional randomized Keynesian beauty contest. Adv. in Appl. Probab. 47 (2015), no. 1, 57--82. doi:10.1239/aap/1427814581. https://projecteuclid.org/euclid.aap/1427814581

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