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Asymptotics of nearly critical Galton-Watson processes with immigration. (English) Zbl 1265.60163
Let $\{\xi_{n,j}, \varepsilon_n: n, j\in {\Bbb N}\}$ be independent nonnegative integer-valued random variables such that $\{\xi_{n,j}: n,j\in {\Bbb N}\}$ are identically distributed. One can define an inhomogeneous Galton-Watson process with immigration $\{X_n: n\in {\Bbb N}\}$ by $X_0 = 0$ and $$ X_n = \sum_{j=1}^{X_{n-1}} \xi_{n,j} + \varepsilon_n. $$ In particular, if $\{\xi_{n,j}: n, j\in {\Bbb N}\}$ follow the Bernoulli distribution, i.e., each particle either dies without descendant or leaves exactly one descendant, one obtains a so-called first-order integer-valued autoregressive (INAR(1)) time series. In that case, the asymptotic behavior of the above process was studied in [{\it L. Györfi} et al., Acta Sci. Math. 73, No. 3--4, 789--815 (2007; Zbl 1174.60042)]. The author of this paper studies similar asymptotics when $\{\xi_{n,j}: n, j\in {\Bbb N}\}$ have a general offspring distribution. He assumes the process is nearly critical, that is $\rho_n := \operatorname{E} \xi_{n,1}\uparrow 1$ as $n\to \infty$. Let $G_n(s) := \operatorname{E}s^{\xi_{n,1}}$ denote the offspring generating function. He shows that, if the second derivative $G_n^{\prime\prime}(1)$ goes to zero rapidly enough, then the asymptotic behaviors are the same as in the INAR(1) case. He also determines the limit if this assumption does not hold showing the optimality of the conditions.

60J80Branching processes