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2019 Analysis of large urn models with local mean-field interactions
Wen Sun, Robert Philippe
Electron. J. Probab. 24: 1-33 (2019). DOI: 10.1214/19-EJP304

Abstract

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time all the balls of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results.

When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

Citation

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Wen Sun. Robert Philippe. "Analysis of large urn models with local mean-field interactions." Electron. J. Probab. 24 1 - 33, 2019. https://doi.org/10.1214/19-EJP304

Information

Received: 5 March 2018; Accepted: 9 April 2019; Published: 2019
First available in Project Euclid: 10 May 2019

zbMATH: 1412.60113
MathSciNet: MR3949270
Digital Object Identifier: 10.1214/19-EJP304

Subjects:
Primary: 60J27, 60K25
Secondary: 68M15

JOURNAL ARTICLE
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