j=0 
= Go€, + Gi€,1+ Gre, 2+--- +Gj Ewjt+--- 
= (GoB° + G\B + G>B* + - ° HG; Bia --) €, 
j=0 
Usually Gp = 1. As the inverse relation to this Green’s function, if 
€,= > (-JjB!) X; 
j=0 
= Clg iB =p Bae : -—1,Bi+- . )X, 
= =NG—Ih G5 Ih G0 > oI Nep—o oo, (G20) 
functions Jp, /;,- - -J ; are called inverse functions, and usually —/p = 1. 
Therefore 
€,=X,-1,X,.-1 —1nX,2-- 9 9) FO. 0 Ones (5.27) 
This inverse function shows how X; can be expressed by AR(©). Therefore the 
inverse function of a pure autoregressive process AR(n) actually has no meaning. For 
example, for AR(1) as in Eq. 5.7; 
Ip =- 1, I, =+a, I; =Oforj = 2. (5.28) 
5.2.2.4 Stationality of AR(1). From Eq. 5.10 
t-1 
X;= lim y, We, = €,+ GE, 1+ a°€,2+- : EE 
FS A) 
Then if 
Ele,] =e for all t, (5.29) 
and as €; represents white noise, 
2 
0. t=u 
Ele;-eoll= sees ; 
[Er €x] 0, bye (5.30) 
we can get 
107 
