arXiv subjects

Thomas Gottfried

Publications and source records attributed to Thomas Gottfried.

3 recordsLinked to original sources

arXiv2024

Tails of explosive birth processes and applications to non-linear P \'{o}lya urns

We derive a simple expression for the tail-asymptotics of an explosive birth process at a fixed observation time conditioned on non-explosion. Using the well-established exponential embedding, we apply this result to compute the tail distribution of the number of balls of a losing colour in generalized P \'{o}lya urn models with super-linear feedback, which are known to exhibit a strong monopoly for the winning colour where losers only win a finite amount. Previous results in this direction were restricted to two colours with the same feedback, which we extend to an arbitrary finite number of colours with individual feedback mechanisms. As an apparent paradox, losing colours with weak feedback are more likely to win in many steps than those with strong feedback. Our approach also allows to characterize the correlations of several losing colours, which provides new insight in the distribution of other related quantities like the total number of balls for losing colours or the time to monopoly. In order to give a complete picture, we also consider sub-linear feedback and discuss asymptotics for a diverging number of colours.

math.PR↗

arXiv2024

Wages and Capital returns in a generalized P\'olya urn

It is a widely observed phenomenon that wealth is distributed significantly more unequal than wages. In this paper we study this phenomenon using a new extension of P\'olyas urn, modelling wealth growth through wages and capital returns. We focus in particular on the role of increasing return rates on capital, which have been identified as a main driver of inequality, and labor share, the second main parameter of our model. We fit the parameters from real-world data in Germany, so that simulation results reproduce the empirical wealth distribution and recent dynamics in Germany quite accurately, and are essentially independent from initial conditions. Our model is simple enough to allow for a detailed mathematical analysis and provides interesting predictions for future developments and on the importance of wages and capital returns for wealth aggregation. We also provide an extensive discussion of the robustness of our results and the plausibility of the main assumptions used in our model.

math.PR↗

arXiv2023

Asymptotics of generalized P\'olya urns with non-linear feedback

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. 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.

math.PR↗