58 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
with 
(83) 
EWdyi—1 Viras )=Bo+B1vi_1 +85 vy, 0+... 
which is a special case of the predictive decomposition of y, 
referred to earlier in this paper [section 1.4 (4)]. The forecasts 
Yn+1> Yn+2 --- are obtained by the chain principle, making iter- 
ated use of (83). Thus when y,,; has been obtained, v,,;., 
is calculated from (83) in the basis of y,.;, v,.; 1, ... The 
variance of the resulting forecasts is given by 
84) 
E(yrumT Ans) = (1 + a? + + a? _,) ok 
showing that the accuracy of the forecast will decrease as the 
forecast span me increases. This last feature is reflected also 
in the JANUS quotient, inasmuch as (84) gives 
I 
(85) E(J)=1+(1- 2) o24 (1-5) a+ + — ab, 
The predictive decomposition (82a-b) has the property that 
the variance (84) is the smallest possible of all representations 
of type (82b). This is the fundamental property of minimum- 
delay, established by E. RoBINsoN, Ref. 24, and referred to 
earlier in this paper. 
The approach (82)-(85) extends to the general stationarv 
case when y, allows the predictive decomposition 
‘86a-b) 
Vi=Y +B ya th 
=W+v,+%, v, +a, 
‘À 
where V, is the deterministic (also called singular) component 
of y,. The procedure of forecasting first settles the prediction 
of the deterministic component over the entire forecast range, 
>] 
Wold - pag. 44