Functional limit theorems for critical processes with immigration

Abstract

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

Approximation of Fluctuations in a Sequence of Nearly Critical Branching Processes

“…The fact that the asymptotic distribution is not normal and the estimator does not have the desired rate of convergence (faster than n −1 ) to the parameter under the estimation are the main causes of the failure. However, as it was shown recently [12] in the process with nonstationary (increasing) immigration, the conditional least-squares estimator of the offspring mean may have normal limit distribution and the rate of convergence is faster than n −1 . This circumstance allows to expect the validity of the bootstrap in this nonclassical model.…”

Validity of the bootstrap in the critical process with a non-stationary immigration

“…We note that when c = 0, the time change in part (c) is ω(t) = t 2+α as in the functional limit theorem for the critical process (Rahimov 2007). The proof of this theorem is based on several preliminary lemmas.…”

“…More detailed discussion and examples can be seen in Rahimov (2009). We also note that the asymptotic distributions for the 'non-weighted' CLSE are obtained in Rahimov (2007). We now describe the bootstrap procedure to approximate the sampling distribution of the pivot…”

Limit theorems of branching processes with immigration

“…Wei 和 Winnicki [17] 证明了临 界分枝过程的阶梯函数在 Skorokhod 拓扑下收敛于一个非负扩散过程. 关于临界分枝过程的泛函极限 定理, 可参见文献 [15][16][17][18][19]. 特别地, Rahimov [18] 证明了当移民趋于无穷且在移民均值和方差满足特定 的关系时, 带独立移民临界分枝过程的泛函中心极限定理的极限过程为 Gauss 过程.…”

“…关于临界分枝过程的泛函极限 定理, 可参见文献 [15][16][17][18][19]. 特别地, Rahimov [18] 证明了当移民趋于无穷且在移民均值和方差满足特定 的关系时, 带独立移民临界分枝过程的泛函中心极限定理的极限过程为 Gauss 过程. Guo 和 Zhang [19] 进一步得到了当各代移民在某种意义下相互依赖时, 过程的一个泛函中心极限定理.…”