具有非线性反馈的广义Pólya瓮的渐近性
Asymptotics of generalized Pólya urns with non-linear feedback
AI总结：
本文系统研究具有非线性反馈的广义Pólya瓮的渐近行为，刻画从次线性到超线性反馈转变中垄断的形成机制，并推导初始市场规模发散极限下的标度极限。
摘要：
具有非线性反馈的广义Pólya瓮是一种成熟的概率模型，用于描述带强化效应的增长过程动力学，一个典型例子是演化市场中智能体之间的竞争。众所周知，反馈机制上的哪些条件会导致垄断，即单个智能体获得全部市场份额，并且针对特定反馈机制的各种进一步结果已从不同角度得到推导。在本文中，我们对一大类一般反馈机制的可能渐近行为给出了全面的阐述，并详细描述了垄断如何通过接近线性的层级状态在从次线性到超线性反馈的转变中涌现。我们进一步区分了超指数和次指数反馈，它们表现出在概念上有趣的差异，有助于理解垄断情形，并研究了渐近性对初始条件、异质性和反馈机制微小变化的稳健性。最后，我们在初始市场规模趋于发散的极限下推导了市场份额完整时间演化的标度极限，包括对典型涨落的描述，并扩展了随机逼近背景下已有的结果。
英文摘要：
Generalized Pólya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement, a generic example being competition of agents in evolving markets. It is well known which conditions on the feedback mechanism lead to monopoly where a single agent achieves full market share, and various further results for particular feedback mechanisms have been derived from different perspectives. In this paper we provide a comprehensive account of the possible asymptotic behaviour for a large general class of feedback, and describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. We further distinguish super- and sub-exponential feedback, which show conceptually interesting differences to understand the monopoly case, and study robustness of the asymptotics with respect to initial conditions, heterogeneities and small changes of the feedback mechanisms. Finally, we derive a scaling limit for the full time evolution of market shares in the limit of diverging initial market size, including the description of typical fluctuations and extending previous results in the context of stochastic approximation.
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