On the survival of a class of subcritical branching processes in random environment

Vincent Bansaye (1) , Vladimir Vatutin (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 SMI | RAS - Steklov Mathematical Institute [Moscow]

Let $Z_{n}$ be the number of individuals in a subcritical BPRE evolving in the environment generated by iid 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 $\rho \in \left( 0,1\right),$ we find the asymptotic survival probability and prove a Yaglom type conditional limit theorem for the process. The survival probability decreases exponentially with an additional polynomial term related to the tail of $X$. The proof relies on 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 $n$ tending to infinity.

Mots clés: Branching processes; random environment; random walk; heavy tails; survival probability

Domaines: Probabilités [math.PR]

Vincent Bansaye, Vladimir Vatutin. On the survival of a class of subcritical branching processes in random environment. 2013. ⟨hal-00844584v2⟩