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Asymptotic results of a multiple-entry reinforcement process

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  • Alves, Caio
  • Ribeiro, Rodrigo
  • Valesin, Daniel

Abstract

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions {Pt}t≥1, where Pt is a partition of {1,2,…,Rt}. At each time t, R numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block’s cardinality, independently. We prove results concerning the asymptotic cardinality of a given block and central limit theorems for associated fluctuations about this asymptotic cardinality: these are proved both for a fixed block and for the maximum among all blocks. We also prove that with probability one, a single block eventually takes and maintains the leadership in cardinality. Depending on the way one sees this partition process, one can translate our results to Balls and Bins processes, Generalized Chinese Restaurant Processes, Generalized Urn models and Preferential attachment random graphs.

Suggested Citation

  • Alves, Caio & Ribeiro, Rodrigo & Valesin, Daniel, 2023. "Asymptotic results of a multiple-entry reinforcement process," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 451-489.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:451-489
    DOI: 10.1016/j.spa.2023.03.010
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    1. Stefano Favaro & Shui Feng & Fuqing Gao, 2018. "Moderate Deviations for Ewens-Pitman Sampling Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 330-341, August.
    2. Caio Alves & Rodrigo Ribeiro & Rémy Sanchis, 2021. "Preferential Attachment Random Graphs with Edge-Step Functions," Journal of Theoretical Probability, Springer, vol. 34(1), pages 438-476, March.
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