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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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