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Poisson limit of an inhomogeneous nearly critical $INAR\left(1\right)$ model. (English) Zbl 1174.60042

A zero start inhomogeneous first order INteger-valued AutoRegressive (INAR(1)) time series ${\left({X}_{n}\right)}_{n\in {ℤ}_{+}}$ is defined as

$\left\{\begin{array}{cc}{X}_{n}=\sum _{j=1}^{{X}_{n-1}}{\xi }_{n,j}+{\epsilon }_{n},\phantom{\rule{1.em}{0ex}}n\in ℕ,\hfill & \\ {X}_{0}=0,\hfill \end{array}\right\$

where $\left\{{\xi }_{n,j},\phantom{\rule{4pt}{0ex}}{\epsilon }_{n}:n,j\in ℕ\right\}$ are independent non-negative integer-valued random variables such that $\left\{{\xi }_{n,j}:j\in ℕ\right\}$ are identically distributed and $P\left({\xi }_{n,1}\in \left\{0,1\right\}\right)=1$ for all $n\in ℕ$. This process can also be written in the form

$\left\{\begin{array}{cc}{X}_{n}={\rho }_{n}\circ {X}_{n-1}+{\epsilon }_{n},\phantom{\rule{1.em}{0ex}}n\in ℕ,\hfill & \\ {X}_{0}=0,\hfill \end{array}\right\$

where ${\rho }_{n}:=\text{E}\phantom{\rule{0.166667em}{0ex}}{\xi }_{n,1}$ and $\circ$ is the so-called Steutel and van Harn operator. One can interpret ${X}_{n}$ as the size of the ${n}^{\text{th}}$ generation of a population, and ${\epsilon }_{n}$ as the number of immigrants in the ${n}^{\text{th}}$ generation. The INAR(1) process ${\left({X}_{n}\right)}_{n\in {ℤ}_{+}}$ is called nearly critical if ${\rho }_{n}\to 1$ as $n\to \infty$.

The authors investigate the asymptotic behavior of nearly critical INAR(1) processes. Under the additional assumption that the factorial moments (specially the mean) of the immigration distributions tend to zero at appropriate speeds they show that the process converges weakly to a Poisson or a compound Poisson distribution. The authors prove their results simultaneously by two different techniques: using probability generating functions and Poisson approximation.

They give an example which shows that the set of possible limiting compound Poisson distributions is not exhausted by their theorems. They pose an open problem that every compound Poisson measure can appear as a limiting distribution of an inhomogeneous INAR(1) process.

##### MSC:
 60J80 Branching processes 60J27 Continuous-time Markov processes on discrete state spaces 60J85 Applications of branching processes