# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Asymptotics of nearly critical Galton-Watson processes with immigration. (English) Zbl 1265.60163

Let $\left\{{\xi }_{n,j},{\epsilon }_{n}:n,j\in ℕ\right\}$ be independent nonnegative integer-valued random variables such that $\left\{{\xi }_{n,j}:n,j\in ℕ\right\}$ are identically distributed. One can define an inhomogeneous Galton-Watson process with immigration $\left\{{X}_{n}:n\in ℕ\right\}$ by ${X}_{0}=0$ and

${X}_{n}=\sum _{j=1}^{{X}_{n-1}}{\xi }_{n,j}+{\epsilon }_{n}·$

In particular, if $\left\{{\xi }_{n,j}:n,j\in ℕ\right\}$ 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 [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 $\left\{{\xi }_{n,j}:n,j\in ℕ\right\}$ have a general offspring distribution. He assumes the process is nearly critical, that is ${\rho }_{n}:=E{\xi }_{n,1}↑1$ as $n\to \infty$. Let ${G}_{n}\left(s\right):=E{s}^{{\xi }_{n,1}}$ denote the offspring generating function. He shows that, if the second derivative ${G}_{n}^{\text{'}\text{'}}\left(1\right)$ 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.

##### MSC:
 60J80 Branching processes