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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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##### References:
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