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Convergence in a multidimensional randomized Keynesian beauty contest.

Grinfeld, Michael and Volkov, Stanislav and Wade, Andrew R. (2015) 'Convergence in a multidimensional randomized Keynesian beauty contest.', Advances in applied probability., 47 (1). pp. 57-82.

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.

Item Type:Article
Keywords:Keynesian beauty contest, Radius of gyration, Rank-driven process, Sum of squared distances.
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1239/aap/1427814581
Date accepted:13 March 2014
Date deposited:26 May 2014
Date of first online publication:31 March 2015
Date first made open access:No date available

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