Asymptotic behaviours of critical branching random walk in $\mathbb{R}^d$

In this paper, we study the asymptotic behaviours of a critical branching random walk in $\mathbb{R}^d$ under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(1,2]$, and that the jump distribution has a finite $\frac{2\alpha}{\alpha-1}$-th moment.

First, we establish the precise decay rate for the tail probability of the all-time maximal displacement $M^d$.

Next, we investigate the maximal displacement $M_n^d$ at generation $n$ and prove a conditional limit theorem for the distribution of $M_n^d$ given that the process survives up to generation $n$.

These results extend the corresponding 1-dimensional results of Lalley and Shao (2015) to the case $d\ge2$.

Finally, we study the asymptotic behaviour of the total progeny $\zeta$.

In particular, we show that, conditioned on the event $\{M^d\ge x\}$, $\zeta$ converges in distribution under an appropriate normalization.

This result reveals a quantitative relationship between the maximal displacement and the total progeny size.