Let {Xn}0∞\documentclass[12pt]{minimal}
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\begin{document}$$\{X_n\}_0^{\infty }$$\end{document} be a supercritical branching process with immigration with offspring distribution {pj}0∞\documentclass[12pt]{minimal}
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\begin{document}$$\{p_j\}_0^{\infty }$$\end{document} and immigration distribution {hi}0∞.\documentclass[12pt]{minimal}
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\begin{document}$$\{h_i\}_0^{\infty }.$$\end{document} Throughout this paper, we assume that p0=0,pj≠1\documentclass[12pt]{minimal}
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\begin{document}$$p_0=0, p_j\ne 1$$\end{document} for any j≥1\documentclass[12pt]{minimal}
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\begin{document}$$j\ge 1$$\end{document} , 1<m=∑j=0∞jpj<∞,\documentclass[12pt]{minimal}
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\begin{document}$$1<m=\sum _{j=0}^{\infty } jp_j<\infty ,$$\end{document} and h0<1\documentclass[12pt]{minimal}
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\begin{document}$$h_0<1$$\end{document}, 0<a=∑j=0∞jhj<∞.\documentclass[12pt]{minimal}
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\begin{document}$$0<a=\sum _{j=0}^{\infty } jh_j<\infty .$$\end{document} We first show that Yn=m-n(Xn-mn+1-1m-1a)\documentclass[12pt]{minimal}
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\begin{document}$$Y_n=m^{-n}(X_n-\frac{m^{n+1}-1}{m-1}a)$$\end{document} is a martingale and converges to a random variable Y. Secondly, we study the rates of convergence to 0 as n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document} of P(Yn-Y>ε),PXn+1Xn-m>ε|Y≥α\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P(\left| Y_n-Y\right|>\varepsilon ), \ \ P\left( \left| \frac{X_{n+1}}{X_n}-m\right| >\varepsilon \Bigg |Y\ge \alpha \right) \end{aligned}$$\end{document}for ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} and α>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >0$$\end{document} under various moment conditions on {pj}0∞\documentclass[12pt]{minimal}
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\begin{document}$$\{p_j\}_0^{\infty }$$\end{document} and {hi}0∞.\documentclass[12pt]{minimal}
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\begin{document}$$\{h_i\}_0^{\infty }.$$\end{document} It is shown that the rates are always supergeometric under a finite moment generating function hypothesis.
