Theoretically, if all the joint distribution functions of all orders 
P1,(%1), Pr(X2), oo 4 
JD KC SO Ma CeO) 2 oo < 
[DeeCSio2eH3S))y 6 0 0 0 
[Din 0 3 o EACSIn heey oo o Beh 00 0900 
are known, the probability structure of X; is completely specified. 
2.1.1 Completely Stationary 
If 
Put 00-0 fr (X1,X2, 0. 950 XP) = Pustkh+k, 09-0 t,+k, ’ Gare: 9.0.0 0 a). (2.3) 
for any f},t2, .... t, and any k, this process is completely stationary. 
2.1.2 Stationary Up to Order m 
In this case, the joint moment up to order m should be the same. 
E[{xcen)]™(XC)}™ [x¢en)} 
= E[Ke1 + HIXea +H"... xe, +0)", (2.4) 
for all positive integers m, m2, ..™, 
where 
mt+m+....m,S™mM. 
2.1.3 Stationary Up to Order 2 
Especially when the order is m = 2, the process is called weakly stationary. When 
the probability density distribution functions are Gaussian, they are completely deter- 
mined by the means, variances, and covariances of two variables, and accordingly they 
are completely stationary. 
2.1.4 Ergodicity 
When the ensemble of the averages across the processes converge to the corre- 
sponding time averages along the process over period N (when N tends toward infinity 
and the mean square is consistent), the process is called ergodic. The ergodicity is a more 
strict condition than the stationality as is shown in Fig. 2.1. 
