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The stationary marginal distribution of a threshold AR(1) process. (English) Zbl 1051.62080
The author considers the threshold AR(1) process defined by $$X_{s}=\tau(X_{s-1})+e_{s}$$, where $\tau(x)=\begin{cases} \alpha x, & \text{if $$x\geq0$$},\\ \beta x, & \text{if $$x<0$$,}\end{cases}$ and $$0<\alpha<1$$ and $$0<\beta<1$$. It is known that under these conditions the process is ergodic. Hence, the stationary marginal distribution exists and is unique. The invariant distribution of this process is computed in the case of the Laplace distributed noise process $$e_{s}$$. The distributions of the threshold AR(1) process for the cases $$0<\alpha<1$$, $$\beta<0$$ and $$0<\alpha<1$$, $$\beta=-\alpha$$ are obtained. The derivation is based on the use of the Frobenius-Perron operator.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E15 Exact distribution theory in statistics
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##### References:
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