×

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)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.2307/2284454 · Zbl 0258.62054 · doi:10.2307/2284454
[2] Anderson O. D., Time Series Analysis and Forecasting-The Box-Jenkins Approach (1976)
[3] Beguin J. M., Time Series pp 423– (1980)
[4] Box G. E. P., Time Series Analysis, 2. ed. (1976) · Zbl 0363.62069
[5] Cox D. R., Scand. J. Statist. 8 pp 93– (1981)
[6] Dossou-Gbete S., Time Series pp 209– (1980)
[7] DOI: 10.2307/2345178 · doi:10.2307/2345178
[8] Granger C. W. J., J. Roy. Statist. Soc. Ser. B 38 pp 189– (1976)
[9] Isserlis L., Biometrika 12 pp 134– (1918) · doi:10.1093/biomet/12.1-2.134
[10] Polya G., Aufgaben und Lehrsatze aus der Analysis 2, 2. ed. (1954) · doi:10.1007/978-3-662-21652-1
[11] DOI: 10.1016/0304-4149(78)90004-2 · Zbl 0387.62074 · doi:10.1016/0304-4149(78)90004-2
[12] Wei W. W. S., Seasonal Analysis of Economic Time Series pp 433– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.