## The stochastic equation $$Y_{n+1}=A_ nY_ n+B_ n$$ with stationary coefficients.(English)Zbl 0588.60056

In this note we deal with the stochastic difference equation of the form $$Y_{n+1}=A_ nY_ n+B_ n$$, $$n\in {\mathbb{Z}}$$, where the sequence $$\Psi =\{(A_ n,B_ n)\}^{\infty}_{n=-\infty}$$ is assumed to be strictly stationary and ergodic. By means of simple arguments a unique stationary solution $$\{y_ n(\Psi)\}^{\infty}_{n=-\infty}$$ 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. $\Psi^ r\to^{{\mathcal D}}_{r\to \infty}\Psi \quad imply\quad \{y_ n(\Psi^ r)\}^{\infty}_{n=-\infty}\to^{{\mathcal D}}_{r\to \infty}\{y_ n(\Psi)\}^{\infty}_{n=-\infty}.$

### MSC:

 60H99 Stochastic analysis 93E15 Stochastic stability in control theory 60F15 Strong limit theorems
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