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The stochastic equation $Y_{t+1}= A_t Y_t+ B_t$ with non-stationary coefficients. (English) Zbl 0988.60034
This paper is concerned with the asymptotics for the sequence $Y_{t+1}=A_t Y_t+B_t$ defined by the non-stationary random environment $(A_t,B_t)_t$. Specifically, convergence in distribution is proven for the shifted process $(Y_{T+t})_t$ as $T \to \infty$ under the condition that $(A_t,B_t)_t$ is stationary under an auxiliary probability which coincides with the original probability on the tail field of $(A_t,B_t)_t$. Moreover, convergence of finite-dimensional marginal distributions is shown to hold true also under the weaker assumption that the process $(A_t,B_t)_t$ can be approximated by certain stationary and ergodic processes.

60G35Signal detection and filtering (stochastic processes)
93E15Stochastic stability
60F99Limit theorems (probability)
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