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be the same in the future as in the past. Thus it is shown that
the observed time series can be represented in the form of a
VR-system, a representation that holds under very general
conditions and to any prescribed accuracy in the stochastic
specification (*2). A similar theorem holds for CC-systems.
Both theorems exist in two versions, one where the observed
set of time series is regarded as a (multidimensional) realization
of a stationary process, and the expectational properties of the
VR- and CC-systems are specified in terms of cross section
averages of the various possible realizations. The other ver-
sion refers to no other realization than the observed time series,
and specifies the expectational properties of the system as
averages over time based on the single realization. The theo-
rems are closely related to the general representation theorem
known as predictive decomposition of stationary stochastic
processes; Refs. 15, 24 and 46. The predictive decomposition
is parametric, and the parameters are uniquely determined. An
important feature of the predictive decomposition and of
CC-systems is that representations of this type yield predictions
that are optimal in the sense of minimum-delay of informa-
tion (°). Thus far we have referred to the given time series as
stationary, but the representation theorems extend to the case
of nonstationary processes; Ref. 25.
Mathematical generalization is not an unmixed blessing.
When a theoretical model is generalized so as to cover wider
areas, the basic assumptions are relaxed to some extent, and
the relaxation brings on that the inference from the model is
attenuated in some respect or other. The ensuing balance be-
tween generalization and attenuation of inference is a most
important aspect of the three models under review. To sum-
marize, any set of observed time series can be cast in the form
(#) For this and the following theorem, see (also for further references)
Ref. 23.
(°) Announced in Ref. 46. the full proof of the CC-representation is as
vet unpublished
2] Wold - pag. 19
