Abstract

Consider a branching random walk on the line. When the branching is critical or subcritical, the system will die out eventually, we will investigate some limit behavior for these cases. 

(1)Conditional CLT for the critical/subcritical branching random walk: Let Zn(B) be the number of  particles located in a Borel set B. We show that under probability P(·|Nn> 0), the proper counting measure Zn(·) converge in law to a random variable, which is specified in terms its moments (critical)/or in terms of the Yaglom limit of the branching processes and the standard Normal varable (subcritical). 

(2)Tail probability of the maximal displacement for subcritical branching random walk: The exact rate for the tail probability of the maximal displacement of subcritical branching random walk is proved, which improve those of Neuman and Zheng (PTRF, 2017). Our method is based on the optional line and spine decomposition. 

This is a joint work with Wenxin Fu, Shengli Liang and Dan Yao.