2022/04/07 by Jean Bertoin, Bertoin, Jean
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2204.03470
We consider a version of the classical Pólya urn scheme which incorporates innovations. The space S of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns C balls of the same color and additional balls of different colors given by some finite point process ξ on S, where the distribution Ps of the pair (C,ξ) depends on the sampled color s. We suppose that the average number of copies Es(C) is the same for all s∈S, and that the intensity measures of innovations have the form Es(ξ)=a(s)μ for some finite measure μ and a modulation function a on S that is bounded away from 0 and ∞. We then show that the empirical distribution of the colors in the urn converges to the normalized intensity ¯¯¯μ. In turn, different regimes for the fluctuations are observed, depending on whether E(C) is larger or smaller than μ(a).