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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 $INAR(p)$ determined by the equation $$X_k=\alpha_1\circ X_{k-1}+\cdots+\alpha_p\circ X_{k-p}+\varepsilon_k,k\in\mathbb N,$$ where $\alpha\circ X=\sum_{j=1}^X\xi_j$ if $X>0$, $\alpha\circ X=0$ if $X>0$, $\xi_j,\ j\in\mathbb N$, are i.i.d. Bernoulli random variables with mean $\alpha\in[0,1]$, $\varepsilon_k$ are i.i.d. non-negative integer-valued random variables with the mean $\mu_{\varepsilon}$, and $\alpha_1,\dots,\alpha_p\in[0,1]$. Under the assumption that the second moment of the innovation $\varepsilon_k$ is finite the authors proved that the sequence of appropriately scaled random step functions $X^n_t=X_{[nt]}/n$, $t\in\mathbb R_+$, $n\in\mathbb N$, formed from an unstable $INAR(p)$ process ($\alpha_1+\cdots+\alpha_p=1$) converges weakly towards a squared Bessel process determined by the stochastic differential equation $dX_t=\left( \mu_{\varepsilon}dt+\sqrt{\sigma^2_{\alpha}X_t^+}dW_t \right)/\varphi'(1)$, $X_0=0$, $t\in\mathbb R_+$, where $\varphi'(1)=\alpha_1+2\alpha_2+\cdots+p\alpha_p>0$, $\sigma^2_{\alpha}=\alpha_1(1-\alpha_1)+2\alpha_2+\cdots+p\alpha_p(1-\alpha_p)$, $W_t,\ t\in\mathbb R_+$, 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 $INAR(p)$ models is quite different from that of familiar (real-valued) unstable autoregressive processes of order $p$ ($AR(p)$ models). An application to Boston armed robberies data is presented.

62M10Time series, auto-correlation, regression, etc. (statistics)
60J80Branching processes
60F99Limit theorems (probability)
62P25Applications of statistics to social sciences
Full Text: DOI arXiv
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