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 $\{\textbf{Z}_n=(Z_n^{(i)})_{1\le i\le d};n\ge0\}$ in both Schr\"{o}der and B\"{o}ttcher cases. We establish explicit decay rates for the following probabilities: $$\mathbb{P}(\textbf{Z}_n=\textbf{k}_n),~ \mathbb{P}(|\textbf{Z}_n|\le k_n), ~\mathbb{P}(Z^{(i)}_n=k_n)~~\text{and}~~\mathbb{P}(Z^{(i)}_n\le k_n), 1\le i \le d,$$ respectively, where $\textbf{k}_n\in\mathbb{Z}_+^d$, $|\textbf{k}_n|=\mathrm{o}(c_n)$, $k_n=\mathrm{o}(c_n)$ and $c_n$ characterizes the growth rate of $\textbf{Z}_n$. These results extend the single-type lower deviation theorems of Fleischmann and Wachtel (Ann. Inst. Henri Poincar\'e Probab. Statist.\textbf{43} (2007) 233-255), thereby paving the way for analysis of precise decay rates of large deviations in multi-type GW processes.