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A unified approach to the study of sums, products, time-aggregation and other functions of ARMA processes. (English) Zbl 0541.62072
Let \(\{X_ t\}\) be a stationary purely non-deterministic process with vanishing mean and with a covariance function r(k). Let B be the backshift operator. The author proves that \(\{X_ t\}\) is ARMA(p,q) iff there exist numbers \(a_ 1,...,a_ p\) such that \(a(z)=1-a_ 1z-...- a_ pz^ p\neq 0\) or \(| z| \leq 1\), \(a(B)r(k)=0\) for \(k>q\), and a(B)r(k)\(\neq 0\) for \(k=q.\)
Using this theorem some conditions are derived, under which sums, products and time-aggregation of ARMA processes follow ARMA processes. The author also considers functions of a Gaussian ARMA process. It is shown that the known results in this field published by other authors can be derived from the main theorem as special cases.
Reviewer: J.Anděl

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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