Large deviation rates for supercritical multitype branching processes with immigration

Let $\{X_n\}_{n\geq0}$ be a $p$-type ($p\geq2$) supercritical branching process with immigration and mean matrix $M$. Suppose that $M$ is positively regular and $\rho$ is the maximal eigenvalue of $M$ with the corresponding left and right eigenvectors $\boldsymbol{v}$ and $\boldsymbol{u}$. Let $\rho>1$ and $Y_n=\rho^{-n}\Big[\boldsymbol{u}\cdot X_n -\frac{\rho^{n+1}-1}{\rho-1}( \boldsymbol{u}\cdot \boldsymbol{\lambda})\Big]$, where the vector $\boldsymbol{\lambda}$ denotes the mean immigration rate. In this paper, we will show that $Y_n$ is a martingale and converges to a $r.v.$ $Y$ as $n\rightarrow\infty$. We study the rates of convergence to $0$ as $n\rightarrow\infty$ of $$ P_i\Big(\Big|\frac{\boldsymbol{l}\cdot X_{n+1}}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot(X_nM)}{\textbf{1}\cdot X_n}\Big|>\varepsilon\Big),P_i\Big(\Big|\frac{\boldsymbol{l}\cdot X_n}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot\boldsymbol{v}}{\textbf{1}\cdot \boldsymbol{v} }\Big|>\varepsilon\Big),P\Big(\Big|Y_n-Y\Big|>\varepsilon\Big) $$ for any $\varepsilon>0, i=1,\cdots,p$, $\textbf{1}=(1,\cdots,1)$ and $\boldsymbol{l}\in\mathbb{R}^p,$ the $p$-dimensional Euclidean space. It is shown that under certain moment conditions, the first two decay geometrically, while conditionally on the event $Y\geq\alpha$ $(\alpha>0)$ supergeometrically. The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.