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Asymptotic behavior of unstable INAR($p$) processes. (English) Zbl 1241.62122

The authors investigate the asymptotic properties of integer-valued autoregressive model $INAR\left(p\right)$ determined by the equation

${X}_{k}={\alpha }_{1}\circ {X}_{k-1}+\cdots +{\alpha }_{p}\circ {X}_{k-p}+{\epsilon }_{k},k\in ℕ,$

where $\alpha \circ X={\sum }_{j=1}^{X}{\xi }_{j}$ if $X>0$, $\alpha \circ X=0$ if $X>0$, ${\xi }_{j},\phantom{\rule{4pt}{0ex}}j\in ℕ$, are i.i.d. Bernoulli random variables with mean $\alpha \in \left[0,1\right]$, ${\epsilon }_{k}$ are i.i.d. non-negative integer-valued random variables with the mean ${\mu }_{\epsilon }$, and ${\alpha }_{1},\cdots ,{\alpha }_{p}\in \left[0,1\right]$. Under the assumption that the second moment of the innovation ${\epsilon }_{k}$ is finite the authors proved that the sequence of appropriately scaled random step functions ${X}_{t}^{n}={X}_{\left[nt\right]}/n$, $t\in {ℝ}_{+}$, $n\in ℕ$, formed from an unstable $INAR\left(p\right)$ process (${\alpha }_{1}+\cdots +{\alpha }_{p}=1$) converges weakly towards a squared Bessel process determined by the stochastic differential equation $d{X}_{t}=\left({\mu }_{\epsilon }dt+\sqrt{{\sigma }_{\alpha }^{2}{X}_{t}^{+}}d{W}_{t}\right)/{\varphi }^{\text{'}}\left(1\right)$, ${X}_{0}=0$, $t\in {ℝ}_{+}$, where ${\varphi }^{\text{'}}\left(1\right)={\alpha }_{1}+2{\alpha }_{2}+\cdots +p{\alpha }_{p}>0$, ${\sigma }_{\alpha }^{2}={\alpha }_{1}\left(1-{\alpha }_{1}\right)+2{\alpha }_{2}+\cdots +p{\alpha }_{p}\left(1-{\alpha }_{p}\right)$, ${W}_{t},\phantom{\rule{4pt}{0ex}}t\in {ℝ}_{+}$, is a standard Wiener process. This limit process is a continuous branching process also known as square-root process or Cox-Ingersoll-Ross process. This asymptotic behavior of unstable $INAR\left(p\right)$ models is quite different from that of familiar (real-valued) unstable autoregressive processes of order $p$ ($AR\left(p\right)$ models). An application to Boston armed robberies data is presented.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 60J80 Branching processes 60F99 Limit theorems (probability) 62P25 Applications of statistics to social sciences