Lower deviation probabilities for supercritical multi-type Galton--Watson processes

This paper provides a detailed analysis of the lower deviation probability properties for a $d$-type ($d>1$) Galton–Watson (GW) process $\{{\mathbf{\text{Z}}}_n=(Z_n^{(i)}{)}_{1\le i\le d};n\ge 0\}$ in both Schröder and Böttcher cases. We establish explicit decay rates for the following probabilities: $$\mathbb{P}({\mathbf{\text{Z}}}_n={\mathbf{\text{k}}}_n),\mathbb{P}(\mathrm{\mid }{\mathbf{\text{Z}}}_n\mathrm{\mid }\le k_n),\mathbb{P}(Z_n^{(i)}=k_n)\text{ and }\mathbb{P}(Z_n^{(i)}\le k_n),1\le i\le d,$$ respectively, where ${\mathbf{\text{k}}}_n\in {\mathbb{Z}}_{+}^d$, $\mathrm{\mid }{\mathbf{\text{k}}}_n\mathrm{\mid }=\mathrm{o}(c_n)$, $k_n=\mathrm{o}(c_n)$ and $c_n$ characterizes the growth rate of ${\mathbf{\text{Z}}}_n$. These results extend the single-type lower deviation theorems of Fleischmann and Wachtel (Ann. Inst. Henri Poincaré Probab. Statist.$\mathbf{\text{43}}$ (2007) 233-255), thereby paving the way for analysis of precise decay rates of large deviations in multi-type GW processes.