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

Abstract

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