TY - JOUR
A1 - Bansaye, Vincent
A1 - BĂ¶inghoff, Christian
T1 - Upper large deviations for branching processes in random environment with heavy tails
T2 - Electronic journal of probability
N2 - ranching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of B\"oinghoff $\&$ Kersting (2009) and Bansaye $\&$ Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.
KW - branching processes
KW - random environment
KW - large deviations
KW - random walks
KW - heavy tails
Y1 - 2011
UR - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/32894
UR - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-328944
SN - 1083-6489
N1 - This work is licensed under a Creative Commons Attribution 3.0 License http://creativecommons.org/licenses/by/3.0/ .
VL - 16
SP - 1900
EP - 1933
PB - EMIS ELibEMS
CY - [Madralin]
ER -