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A note on the threshold AR(1) model with Cauchy innovations. (English) Zbl 0587.60033
The paper investigates the autoregressive process \(\{X_ s\}\) of first order with one threshold \(r=0\) given by the following formulae \[ (1)\quad X_ s=\alpha X_{s-1}+\ell_ s\quad if\quad X_{s-1}\leq r,\quad X_ s=\beta X_{s-1}+\ell_ s\quad if\quad X_{s-1}>r \] where \(\alpha =-\beta\), \(\alpha\in (0,1)\) and \(\ell_ 1,...,\ell_ n,..\). are independent C(0,1) Cauchy variables with density \(g(x)=\pi^{-1}\{1+x^ 2\}^{-1}\), \(-\infty <x<+\infty\). The authors have found an exact solution for the stationary density h(x) of the distribution of \(X_ s\) of the threshold autoregressive process (1) with Cauchy innovations.
Reviewer: I.G.Zhurbenko

MSC:
60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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References:
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[2] DOI: 10.2307/3213639 · Zbl 0541.62073 · doi:10.2307/3213639
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