Me a1 
E[X,] = lim Me [I1ta+a’+: +a} = lim l—a » (5.31) 
oe eo) 1 © 
Me t a=1 
t-1 : t-1 
cov. [X,;-X,4,] = lim ») ae, _; Sy GE rs pj 
be i=0 j=0 
= lim ota’ +a"? + rar e2e-o} : 
to 
1 2t 
oa’ 2 ail 
Therefore R(r) = lim ie : (5.32) 
1 © 
o2t, a=1 
If ue = 0, 
1—a~ 
oz => @ #1 
var. [XJ=R0)=lim4 1-4 (5.33) 
t— © 
o2t, a=1 
From these results: 
[a.] When. = 0 
E[X,] is, from Eq. 5.31, a function of time and therefore {X;} 1s not stationary, even 
to order 1. 
(1.) When lal < 1, and when t tends to a large value 
E[X,;] = = = const. by ¢. (5.34) 
Therefore {X,} is asymptotically stationary to order 1. 
(2.) When a= 1 
E[X;] =Met (5.35) 
E|X] increases by time ¢. Therefore, {x,| is not stationary, even up to order 1. This is the 
case of Random Walk. 
[ b.] When, =0 
E[X,] = 0. 
cov. [X;- Xr] = Rr) = 2a" aE 
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