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Linear random fields. (English) Zbl 0594.60052

Studies in econometrics, time series, and multivariate statistics, Commem. T. W. Anderson’s 65th Birthday, 299-309 (1983).
[For the entire collection see Zbl 0528.00008.]
Let \(Z^ k\) be the set of k-vectors with integer components. Consider an additive semigroup S that is a subset of \(Z^ k\) with 0 an element. Let \(V_ t\) be the ”white noise” field on \(Z^ k\), i.e. E \(V_ t=0\), E \(V_ tV_{\tau}=\delta_{t-\tau}\). A weakly stationary solution \(X_ t\) of the system equations \[ \sum_{\tau \in S}b_{\tau}X_{t+\tau}=\sum_{\tau \in S}a_{\tau}V_{t+\tau} \] is called an ARMA field relative to the semigroup S. Conditions under which ARMA fields exist, are studied. The problem of linear prediction of \(X_ 0\) in terms of \(X_{\tau}\), \(\tau\in S\setminus \{0\}\) is considered. Some properties of linear random fields \[ X_ t=\sum_{\tau}d_{\tau}V_{t-\tau} \] where \(V_ t\) is a sequence of independent identically distributed random variables, are investigated.
Reviewer: M.Yadrenko

MSC:

60G60 Random fields
60G25 Prediction theory (aspects of stochastic processes)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Citations:

Zbl 0528.00008