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The stochastic equation $Y\sb{n+1}=A\sb nY\sb n+B\sb n$ with stationary coefficients. (English) Zbl 0588.60056
In this note we deal with the stochastic difference equation of the form $Y\sb{n+1}=A\sb nY\sb n+B\sb n$, $n\in {\bbfZ}$, where the sequence $\Psi =\{(A\sb n,B\sb n)\}\sp{\infty}\sb{n=-\infty}$ is assumed to be strictly stationary and ergodic. By means of simple arguments a unique stationary solution $\{y\sb n(\Psi)\}\sp{\infty}\sb{n=-\infty}$ of this equation is constructed. The stability of the stationary solution is the second subject of investigation. It is shown that under some additional assumptions. $$ \Psi\sp r\to\sp{{\cal D}}\sb{r\to \infty}\Psi \quad imply\quad \{y\sb n(\Psi\sp r)\}\sp{\infty}\sb{n=-\infty}\to\sp{{\cal D}}\sb{r\to \infty}\{y\sb n(\Psi)\}\sp{\infty}\sb{n=-\infty}.$$

60H99Stochastic analysis
93E15Stochastic stability
60F15Strong limit theorems
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