Electronic Journal of Probability

Scaling limits of population and evolution processes in random environment

Vincent Bansaye, Maria-Emilia Caballero, and Sylvie Méléard

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We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 19, 38 pp.

Received: 19 February 2018
Accepted: 2 January 2019
First available in Project Euclid: 8 March 2019

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Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes 60F15: Strong theorems 60F05: Central limit and other weak theorems 60F10: Large deviations 92D25: Population dynamics (general)

tightness diffusions with jumps characteristics semimartingales Galton-Watson process Wright-Fisher process random environment

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Bansaye, Vincent; Caballero, Maria-Emilia; Méléard, Sylvie. Scaling limits of population and evolution processes in random environment. Electron. J. Probab. 24 (2019), paper no. 19, 38 pp. doi:10.1214/19-EJP262. https://projecteuclid.org/euclid.ejp/1552013626

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