Large deviation rates for supercritical multitype branching processes with immigration

TL;DR

This paper investigates the convergence rates of certain probabilities related to supercritical multitype branching processes with immigration, showing geometric decay under moment conditions and supergeometric decay under stronger assumptions.

Contribution

It establishes the decay rates of probabilities involving the process and its martingale limit, providing new insights into their convergence behavior under various moment conditions.

Findings

First two probabilities decay geometrically under moment conditions.

Conditional decay of the third probability is supergeometric.

Supergeometric decay always occurs under finite moment generating function assumption.

Abstract

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.