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On stationarity of the solution of a doubly stochastic model. (English) Zbl 0595.60041
The paper presents a study of a stochastic process $$X_ t$$, $$t=...- 1,0,1,..$$. defined by the difference equation $X_ t=\phi_ tX_{t- 1}+\epsilon_ t$ where $$\epsilon_ t$$ is a sequence of independent identically distributed random variables and $$\phi_ t$$ is a stochastic process defined on the probability space of $$\{\epsilon_ t\}.$$
Necessary and sufficient conditions on $$\{\phi_ t\}$$ are given for $$\{X_ t\}$$ to be a doubly stochastic moving average process $$X_ t=\sum_{k}\beta_ k\epsilon_{t-k},$$ where $$\{\beta_ t\}$$ is a stochastic process in $$L^ 2$$, or for $$\{X_ t\}$$ to be a second order strictly stationary process.
Reviewer: I.G.Zhurbenko

##### MSC:
 60G10 Stationary stochastic processes 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60E10 Characteristic functions; other transforms
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