Let {Xn}n≥0 be a p-type (p ≥ 2) supercritical branching process with immigration and mean matrix M. Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u. Let ρ > 1 and \(Y_{n}=\rho^{-n}\left[{\bf u}\cdot{X}_{n}-{{{\rho}^{n+1}-1} \over {\rho}-1}\left({\boldsymbol u} \cdot {\boldsymbol \lambda}\right)\right]\) , where the vector λ denotes the mean immigration rate. In this paper, we will show that Yn is a martingale and converges to an r.v. Y as n → ∞. We study the rates of convergence to 0 as n → ∞ of \({P}_{i}\left(\left\vert{{\boldsymbol l}\cdot{X}_{{n}+1} \over {\bf 1}\cdot{X}_{n}} - {{{\boldsymbol l}\cdot({X}_{n}M)} \over {\bf 1}\cdot{X}_{n}} \right\vert > \varepsilon \right),\quad {P}_{i}\left(\left\vert{{\boldsymbol l}\cdot{X}_{{n}} \over {\bf 1}\cdot{X}_{n}} - {{{\boldsymbol l}\cdot{\boldsymbol v}} \over {\bf 1}\cdot{\boldsymbol v}} \right\vert > \varepsilon \right),\quad P(\vert{Y}_{n} - {Y}\vert > \varepsilon)\) for any ε > 0, i = 1,…,p, 1 = (1,…,1) and l ∈ ℝp, the p-dimensional Euclidean space. It is shown that under certain moment conditions, the first two decay geometrically, while conditionally on the event Y ≥ α (α > 0) supergeometrically. The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.
