Asymptotics of generalized P\'olya urns with non-linear feedback
| Autor: | Gottfried, Thomas, Grosskinsky, Stefan |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Generalized P\'olya 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. Depending on the feedback function, it is well known that the model may exhibit monopoly, where a single agent achieves full market share. Besides this general result, various further results for particular feedback mechanisms have been derived from different perspectives. The purpose of this paper is to provide a comprehensive account of the possible asymptotic behaviour for a large general class of feedback functions. We distinguish two main types of feedback, (super-)exponential or sub-exponential, that require separate approaches to understand the monopoly case. This difference is also conceptually interesting, and we present a general criterion to predict the (in general random) monopolist with high probability under generic initial conditions. In the non-monopoly case, we establish the long-time asymptotics of market shares, which even holds when there is no stable limit. We also include a detailed analysis of almost linear feedback functions to characterize the transition between non-monopoly and monoply. Finally, we derive a scaling limit for the full time evolution of market shares in the limit of diverging initial market size, including a description of typical fluctuations and extending previous results in the context of stochastic approximation. Comment: 59 pages, 7 Figures |
| Databáze: | arXiv |
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