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Random coefficient autoregression regime switching and long memory. (English) Zbl 1044.62095

Let \(X_t=a_t X_{t-1}+\varepsilon_t\) be a random coefficient AR model where \(\{\varepsilon_t\}\) are iid (0,1) random variables. The authors assume that \(\{a_t\}\) is the renewal-reward process such that \(a_t=A_j\) for \(S_{j-1}<t\leq S_j\) \((j\in\mathbb Z)\), \(\{S_j\}\) is a stationary renewal process on \(\mathbb Z\) and \(\{A_j\}\) is a sequence of iid random variables. Let the inter-renewal times \(U_i=S_i-S_{i-1}\) be iid random variables with finite expectation. Necessary and sufficient conditions for the existence of a covariance stationary solution \(X_t = \sum_{i=0}^{\infty} \varepsilon_{t-i} \prod_{p=0}^{i-1} a_{t-p}\) are derived. If the marginal distribution of \(A_i\) has either an atom at \(a=1\) or a beta-type density in a neighborhood of \(a=1\), then it is shown that the covariance function of \(\{X_t\}\) decays hyperbolically, and normalized partial sums of the process \(\{X_t\}\) converge weakly to a \(\lambda\)-stable Lévy process.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60K05 Renewal theory
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
60F05 Central limit and other weak theorems
60G52 Stable stochastic processes
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