论文标题
限制与移民的连续分支过程的定理
Limit theorems for continuous-state branching processes with immigration
论文作者
论文摘要
我们证明并扩展了Mark Pinsky所说的一些结果：限制了与移民的连续状态分支过程的定理[Bull。阿米尔。数学。 Soc。 78（1972），242--244]。考虑一个带有移民$（y_t，t \ geq 0）$的连续国家分支过程，带有分支机构$ψ$和移民机制$φ$（CBI $（ψ，φ）$，短）。当$ \ int_ {0} \ frac {φ（u）} {|ψ（u）|} du <\ infty $或$ \ int_ {0} {0} {0} \ frac {φ（U）{φ（U）} {U）{U）{U）{U）{U）{U）{我们首先观察到，当$ \ int_ {0} \ frac {φ（u）} {|ψ（u）|} du <\ infty $时，超临界CBI的增长率是由分支动力学决定的，即重新分配$τ（t）$，仅$ convent $ convery（t）几乎是有限的随机变量。当$ \ int_ {0} \ frac {φ（u）} {|ψ（u）|} du = \ infty $时，这表明移民淹没了分支动力学，并且该过程不存在线性肾均值化。通过法律中的非线性时间依赖性重归于所有非关键CBI过程的详细信息，研究了第二条制度中的渐近学。然后展示了三个弱收敛性的制度，其中纠正了Pinsky论文中的错误。还研究了具有定期变化假设的关键分支机制的CBI过程。
We prove and extend some results stated by Mark Pinsky: Limit theorems for continuous state branching processes with immigration [Bull. Amer. Math. Soc. 78(1972), 242--244]. Consider a continuous-state branching process with immigration $(Y_t,t\geq 0)$ with branching mechanism $Ψ$ and immigration mechanism $Φ$ (CBI$(Ψ,Φ)$ for short). We shed some light on two different asymptotic regimes occurring when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$ or $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$. We first observe that when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$, supercritical CBIs have a growth rate dictated by the branching dynamics, namely there is a renormalization $τ(t)$, only depending on $Ψ$, such that $(τ(t)Y_t,t\geq 0)$ converges almost-surely to a finite random variable. When $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$, it is shown that the immigration overwhelms the branching dynamics and that no linear renormalization of the process can exist. Asymptotics in the second regime are studied in details for all non-critical CBI processes via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are then exhibited, where a misprint in Pinsky's paper is corrected. CBI processes with critical branching mechanisms subject to a regular variation assumption are also studied.
