(1.) When lal < 1, i.e., when {Z) = Z—a = 0 has its root inside the unit circle or 
when @(Z) = 1—aZ = 0 has its root outside the unit circle, from Eq. 5.32, even E{X;} = 0 
holds, the cov. [X,;, X, 
up to order 2. However, when t—> & , 
is a function of time. Accordingly, {X;} is not stationary even 
+7] 
2 
0. 
cov. [X,-X,,,] =RW)= i 5 a’ const. by t, (5.36) 
-—a 
a2 
ar. [X;] = E[X, X,] = R(O) = const. by 1. (5.37) 
-—a 
Therefore, {X;} is asymptotically stationary up to order 2. The form of R(r) will be 
different by the sign of a, although it gradually decays as increases as in Fig. 5.8. 
Sale 
(a)O0<a<1 (b) -1<a<0 
Fig. 5.8. A(A of AR(1). 
(2.) When lal > 1, i.e., when f(Z) = Z—a has its root outside the unit circle, or when 
a(Z) = 1 —aZ has its root inside the unit circle, 
2 
cov. [X; - Xu,] = R(r) = = i a’(a—1) (5.38) 
oO 2 
var. [X= Ai GEN) (5.39) 
a — 
In this case, E[X] = 0, but not only does R(r) not converge to a small value as r increases, 
but both R(r) and R(O) change by the time ¢ and are not stationary. As time passes, these 
values continue to increase. Accordingly, this process is not stationary up to order 2, even 
asymptotically. 
5.2.2.5 Autocovariance and Spectrum of AR(1). The general solution of R(r) can 
also be obtained more simply as the solution of a first order homogenous difference equa- 
tion, because in this case, from Eq. 5.7, 
X, = aX;1+€1, (3.7) 
assuming 4“ « = 0 and E[X,] = 0. Multiplying both sides of Eq. 5.7 by X,_, and taking the 
expected values gives 
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