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On random coefficient INAR(1) processes. (English) Zbl 1271.60094

First-order random coefficient integer-valued autoregressive (abbreviated as RCINAR(1)) process is the following recursion \[ X_n=\sum_{k=1}^{X_{n-1}} B_{n,k}+Z_n, \quad n\in\mathbb{N}, \] where \(X_0=0\), \(\{Z_n\}\) is a sequence of i.i.d. integer-valued nonnegative random variables, \(\{B_{n,k}\}\) is a collection of independent Bernoulli random variables such that \[ \operatorname{P}\{B_{n,k}=1\}=\phi_n, \quad \operatorname{P}\{B_{n,k}=0\}=1-\phi_n \quad \text{for all } k \in \mathbb{N}, \] with \(\{\phi_n\}\) independent i.i.d. real-valued random variables such that \(\phi_n \in [0,1]\). It is assumed that \(\{Z_n\}\), \(\{B_{n,k}\}\), \(\{\phi_n\}\) are independent of each other.
Formally, the RCINAR(1) can be classified as the branching process with immigration in a random environment \(\{\phi_n\}\) as follows. At the beginning of the \(n\)-th period, \(Z_n\) immigrants enter the system and each particle from the previous generation \(X_{n-1}\) can produce one child with probability \(\phi_n\) or none with probability \(1-\phi_n\).
In the paper, the asymptotic behavior of the model is considered in the case where \(\{Z_n\}\) belongs to the domain of attraction of a stable law. In particular, weak limits of extreme values and the growth rate of partial sums are studied.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K37 Processes in random environments
60F05 Central limit and other weak theorems
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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