Electronic Journal of Probability

On the scaling limits of Galton-Watson processes in varying environments

Vincent Bansaye and Florian Simatos

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Abstract

;We establish a general sufficient condition for a sequence of Galton-Watson branching processes in varying environments to converge weakly. This condition extends previous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real-valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 75, 36 pp.

Dates
Accepted: 3 July 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067181

Digital Object Identifier
doi:10.1214/EJP.v20-3812

Mathematical Reviews number (MathSciNet)
MR3371434

Zentralblatt MATH identifier
1321.60171

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments

Keywords
branching process varying environment scaling limit

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bansaye, Vincent; Simatos, Florian. On the scaling limits of Galton-Watson processes in varying environments. Electron. J. Probab. 20 (2015), paper no. 75, 36 pp. doi:10.1214/EJP.v20-3812. https://projecteuclid.org/euclid.ejp/1465067181


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