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The Stochastic Equation Yn+1=AnYn+Bn with Stationary Coefficients

Andreas Brandt
Advances in Applied Probability
Vol. 18, No. 1 (Mar., 1986), pp. 211-220
DOI: 10.2307/1427243
Stable URL: http://www.jstor.org/stable/1427243
Page Count: 10
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The Stochastic Equation Yn+1=AnYn+Bn with Stationary Coefficients
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Abstract

In this note we deal with the stochastic difference equation of the form Yn+1=AnYn+Bn,n∈ Z, where the sequence Ψ ={(An,Bn)}n=-∞ ∞ is assumed to be strictly stationary and ergodic. By means of simple arguments a unique stationary solution {yn(Ψ)}n=-∞ ∞ 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 $\Psi ^{r}\underset r\rightarrow \infty \to{\overset \scr{D}\to{\rightarrow}}\Psi \ \text{imply}\ \{y_{n}(\Psi ^{r})\}_{n=-\infty}^{\infty}\underset r\rightarrow \infty \to{\overset \scr{D}\to{\rightarrow}}\{y_{n}(\Psi)\}_{n=-\infty}^{\infty}$.

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