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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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