Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.3 ERGODICITY

Consider first ergodicity in the mean. A time average for the mean of random process X(t) is given in (8.8). We know that if X(t) has a constant mean, then : the expected value of the time average, even for finite T, is the mean of the random process so that is an unbiased estimator of μ X. Ergodicity in the mean is a stronger property such that the time average itself converges to the mean as .

Definition: Ergodic in Mean (Process) Random process X(t) with constant mean μ X is ergodic in the mean if

(8.14)

When the time average approaches its expectation as implied above, its variance approaches zero. This is a very useful property: the mean of a random process can be accurately estimated by averaging any realization over a sufficiently long time interval. Ergodicity in the mean is verified as follows.

Theorem 8.1 (Ergodic in Mean). Random process X(t) with autocovariance function CXX(τ) is ergodic in the mean if and only if

(8.15)

where

(8.16)

Proof. Consider the ...