超临界分枝过程的大偏差速率

Large Deviation Rates for Supercritical Branching Process

假设{Y(t);t ≥0}是带移民的连续时间分枝过程,其中分枝概率是{bk;k≥0},移民概率是{aj;j≥0}。令b0=0,0 k≠1(k≥1),1 k=0∞kbk j=0∞jbj 。首先,我们证明 是一个上鞅并且收敛到随机变量K。然后,我们在α 】0和ε 】0时,当{bk;k≥0}和{ak;k≥0}满足多种矩条件,研究P(|K(t)-K| 】ε)在t趋于无穷时的衰减速率。

Suppose {Y(t);t ≥0} is the continuous time supercritical branching process with offspring rates {bk;k≥0} and immigration rates {aj;j≥0}. Let b0=0, 0 k≠1(k≥1),1 k=0∞ kbk j=0∞ jbj . Firstly, we suppose that is a sub-martingale and converges to a random variable K. Then we study the decay rates of P(|K(t)-K| >ε) as t→∞ for α >0, ε >0 under various moment conditions on {ak;k≥0} and {bk;k≥0}.