2.2.4 Consistency 
As the number of the samples n tends toward infinity, var. (6) — 0, and bias 
b2(6) — 0 must be satisfied. Then the mean square error M2(6) — 0 also stands and is 
called the “mean square consistency.” 
225 Sufficiency 
The estimator @ must contain all the information X;,X2, . .. . X, in the sample, 
relevant to the estimator of 6 
(30 Cano, sacs SO 
2.3 AUTOCOVARIANCE FUNCTION AND ITS ESTIMATES 
When the process X(t) is stationary up to an order of 2, the covariance function 
cov. [X(t) X(t+7)] = E [|X(t)—pl{Xr+7)-u}] = R@ (2.18) 
is a function oft only. Then, 0(t) = R(t)/R(O) is called an autocovariance coefficient, 
and 
RO) = E [{X()—-pw}*] = var. [X()] = 07. (2.19) 
When the process is real valued, R(—T) = R(t) as in Fig. 2.2, and when the process is 
complex valued, R(—T) = R*(t). This function R(z) is a measure of similarity in a sense, 
and is also a measure of the memory of the process. R(t) plays a big role later, in the 
parametric analysis in Part II, in identifying the statistical model that will fit the process. 
Meany, varianceo*, R(t), and o(t) are constant by t. 
{ R(t) 
Fig. 2.2. Autocovariance function. 
10 
