The generalized Pòlya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the urn model in practice. Specifically, we establish the law of large numbers and a central limit theorem for the number of sampled balls. To establish these results, we develop new techniques involving last exit times and crossing time analyses of the proportion of balls in the urn. To obtain precise estimates in these techniques, we establish results on the harmonic moments of the total number of balls in the urn. Finally, we describe our main results in the context an application to response-adaptive randomization in clinical trials. Our simulation experiments in this context demonstrate the ease and scope of our model.
Synchronization of Reinforced Stochastic Processes with a Network-based Interaction
由相互作用的元素组成的随机演化系统一直是许多科学领域的研究热点。这项工作涉及的同步现象，可以大致定义为不同的组件采取共同的行为的趋势。我们继续研究具有强化的相互作用随机过程的模型，该模型最近在Crimaldi et al.（2016，arXiv：1602.06217）中引入。一般来说，我们所说的强化是指这样一种机制，即给定事件发生的概率越来越依赖于同一类型事件在过去发生的次数。这种随机过程系统的特殊性在于，同步是由强化机制本身沿时间诱导的，不需要大规模的限制。我们重点研究了相互作用网络的拓扑结构与长时间同步现象之间的关系。在证明了几乎必然同步性之后，我们给出了稳定收敛意义下的CLT，建立了收敛到公共极限和同步的收敛速度和渐近分布.所得到的结果导致建设的渐近置信区间的限制随机变量和统计测试作出推断的拓扑结构的网络给定的观察加强随机过程定位在顶点。
