Open Access
February 2017 On the survival probability for a class of subcritical branching processes in random environment
Vincent Bansaye, Vladimir Vatutin
Bernoulli 23(1): 58-88 (February 2017). DOI: 10.3150/15-BEJ723


Let $Z_{n}$ be the number of individuals in a subcritical Branching Process in Random Environment (BPRE) evolving in the environment generated by i.i.d. probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of $X$ has the form

\[p_{X}(x)=x^{-\beta-1}l_{0}(x)e^{-\rho x}\] for some $\beta>2$, a slowly varying function $l_{0}(x)$ and $\rho\in(0,1)$, we find the asymptotic of the survival probability $\mathbb{P}(Z_{n}>0)$ as $n\rightarrow\infty$, prove a Yaglom type conditional limit theorem for the process and describe the conditioned environment. The survival probability decreases exponentially with an additional polynomial term related to the tail of $X$. The proof uses in particular a fine study of a random walk (with negative drift and heavy tails) conditioned to stay positive until time $n$ and to have a small positive value at time $n$, with $n\rightarrow\infty$.


Download Citation

Vincent Bansaye. Vladimir Vatutin. "On the survival probability for a class of subcritical branching processes in random environment." Bernoulli 23 (1) 58 - 88, February 2017.


Received: 1 December 2013; Revised: 1 March 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1362.60074
MathSciNet: MR3556766
Digital Object Identifier: 10.3150/15-BEJ723

Keywords: branching processes , heavy tails , random environment , Random walks , speed of extinction

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
Back to Top