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

Brandt, Andreas

Adv. Appl. Probab. 18, 211-220 (1986).

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

Keywords:

uniform strong law of large numbers; stochastic difference equation; strictly stationary and ergodic; stationary solution