Gaussian fluctuations for the two urn model

2023/01/20 by Kolesko, Konrad, Sava-Huss, Ecaterina
#60B20 #60F05 #60F17 #60J80 #60J85 #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2301.08602

Abstract

We introduce a modification of the generalized Pólya urn model containing two urns and we study the number of balls B_j(n) of a given color j∈\1,…,J\, J∈ℕ added to the urns after n draws. We provide sufficient conditions under which the random variables (B_j(n))_n∈ℕ properly normalized and centered converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second order behavior occurs at √ρ and not at ρ/2, where ρ is the dominant eigenvalue of the mean replacement matrix.

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