March 2023 Scaling limits and fluctuations of a family of N-urn branching processes
Lirong Ren, Xiaofeng Xue
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Braz. J. Probab. Stat. 37(1): 195-218 (March 2023). DOI: 10.1214/23-BJPS567

Abstract

In this paper, we are concerned with a family of N-urn branching processes, where some particles are initially placed in N urns, and then each particle gives birth to several new particles in some urn when dies. This model includes the N-urn Ehrenfest model and N-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a C(T)-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein–Uhlenbeck process in the dual of C(T), where T=(0,1] is the one-dimensional torus. A crucial step for the proofs of the above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, the limit theorems of the hitting times of the process are also discussed.

Funding Statement

The second author was supported by Fundamental Research Funds for the Central Universities with grant number 2022JBMC039.

Acknowledgments

The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Lirong Ren. Xiaofeng Xue. "Scaling limits and fluctuations of a family of N-urn branching processes." Braz. J. Probab. Stat. 37 (1) 195 - 218, March 2023. https://doi.org/10.1214/23-BJPS567

Information

Received: 1 October 2022; Accepted: 1 February 2023; Published: March 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580891
zbMATH: 1511.60129
Digital Object Identifier: 10.1214/23-BJPS567

Keywords: branching process , Fluctuation , Scaling limit

Rights: Copyright © 2023 Brazilian Statistical Association

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Vol.37 • No. 1 • March 2023
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