Asymptotic Results of a Multiple-entry Reinforcement Process

TL;DR

This paper studies a class of reinforced stochastic processes involving random partitions, proving asymptotic behaviors, fluctuations, and leadership properties, with applications to various models like urns, Chinese restaurant processes, and preferential attachment graphs.

Contribution

It introduces a new class of reinforcement processes with detailed asymptotic and fluctuation results, extending understanding of block growth and leadership in complex stochastic models.

Findings

Asymptotic cardinality of blocks is established.

Central limit theorems for fluctuations are proved.

A single block eventually dominates in size with probability one.

Abstract

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. 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.