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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vet unpublished 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