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From Gini index as a Lyapunov functional to convergence in Wasserstein distance

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  • Fei Cao

Abstract

In several recent works on infinite-dimensional systems of ODEs \cite{cao_derivation_2021,cao_explicit_2021,cao_iterative_2024,cao_sticky_2024}, which arise from the mean-field limit of agent-based models in economics and social sciences and model the evolution of probability distributions (on the set of non-negative integers), it is often shown that the Gini index serves as a natural Lyapunov functional along the solution to a given system. Furthermore, the Gini index converges to that of the equilibrium distribution. However, it is not immediately clear whether this convergence at the level of the Gini index implies convergence in the sense of probability distributions or even stronger notions of convergence. In this paper, we prove several results in this direction, highlighting the interplay between the Gini index and other popular metrics, such as the Wasserstein distance and the usual $\ell^p$ distance, which are used to quantify the closeness of probability distributions.

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  • Fei Cao, 2024. "From Gini index as a Lyapunov functional to convergence in Wasserstein distance," Papers 2409.15225, arXiv.org.
    • Handle: RePEc:arx:papers:2409.15225
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    1. Boghosian, Bruce M. & Devitt-Lee, Adrian & Johnson, Merek & Li, Jie & Marcq, Jeremy A. & Wang, Hongyan, 2017. "Oligarchy as a phase transition: The effect of wealth-attained advantage in a Fokker–Planck description of asset exchange," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 15-37.
    2. Adrian Dragulescu & Victor M. Yakovenko, 2000. "Statistical mechanics of money," Papers cond-mat/0001432, arXiv.org, revised Aug 2000.
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