Mathematics – Probability
Scientific paper
2011-12-22
Mathematics
Probability
Scientific paper
Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive values, but lower than expected. First, we consider small positive values the process may reach for large times and describe the asymptotic behavior of $\P(1 \leq Z_n \leq k)$ as $n\rightarrow \infty$. If the reproduction laws are linear fractional, two regimes appear for the rate of decrease of this probability. Secondly, we are interested in the lower large deviations of $Z$ and give the rate function under some moment assumptions. This result generalizes the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where $\P_1(Z_1=0)>0$ but also weaker moment assumptions.
Bansaye Vincent
Boeinghoff Christian
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