© 1990 by Biometrika Trust
AMENDMENTS AND CORRECTIONS |
‘The uniqueness of moving average representations with independent and Identically distributed random variables for non-Gaussian stationary time series’
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Abstract |
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Formula (1·5) shows that the uniqueness result holds under less restrictive assumptions than were given in the Theorem. If the rth cumulant of xt, exists and is nonzero for some r
3, then the vanishing of (1·5) contradicts nonuniqueness (1·4). Higher than rth order moments and full stationarity are not required, only stationarity of moments up through order r. Independence of the 
t is not required, only the vanishing of cumulants of the form cum (
) with s 
r whenever at least two of the indices t,...,ts, are distinct.
The author was stimulated to make these remarks by reading a diflEerent proof of the uniqueness result in an unpublished Peking University report by Q. Cheng, who assumed full stationarity and independence of the t, but only rth order moments. Cheng also asserts that the analogous uniqueness result holds for spatial series with j = (k, l) k, l=0,±1,...). In fact, all that is required is that the indices j range over a countable, commutative group J. In (1·5), d
. is then to be interpreted as Haar measure on the character group of J, and eij
is a symbol denoting the result of applying the character to j (Blanc-Lapierre & Fortet, 1968, pp. 268–75; Rudin, 1962, pp. 1–27).