then XOS y BE (5.10) 
or, considering the stationarity of X, and assuming that X_, exist, 
t 
xXGS » a'~“e,+a*NX 4. If X_n is 0, when N tends to 
u=~(N-1) 
t oe) 
GS SES ae (5.11) 
j=0 
u=-@ 
which is the same form as Eq. 5.10. Setting 
Gj=ai (5.12) 
gives 
[--) t 
X;= Sy Gj erj= Sy, G,.j €j- 638) 
j=0 
F=—0O 
G; is called Green’s function for AR(1). 
Green’s function G; shows the weight given in the present response of X; to the dis- 
turbance €,_;, which entered the system j time units back. It also indicates how well the 
system remembers the disturbance €,; or how slowly or quickly the dynamic response of 
the system to any particular €,_; decays. Equation 5.13 is called Wold’s decomposition 
and gives the decomposition of X; into an infinite number of orthogonal variables G; €,_; . 
Equation 5.13 also implies that the AR(1) process can be inverted into an infinite order 
moving average process MA (~) as 
Xe = Ge; Gi €ni+ G6 fot G ee Ee Galsy) 
5.22.2 Solution of the Difference Equation. Equation 5.7 is a difference equation 
and can be written 
X,— Xx} =€;. G79) 
The general solution of the difference equation is the sum of the solution of its 
homogeneous equation plus a particular solution of Eq. 5.7’. The homogeneous 
equation of Eq. 5.7’ is 
X;—aX,; = 0, (5.14) 
its solution being 
Xo—An (5.15) 
where yw is the root of its characteristic equation, equated to zero. 
105 
