Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.  ## Ergodicity

When we look at the thermal noise of a resistor, we find that the noisy signal fluctuates around zero leading to a mean over time of zero. The mean over time does not depend on the specific point of time when it is determined, the signal is said to be stationary.

If we take several resistors (having all the same resistance) and measure the noise of each resistor at the same time, we again find that the mean of the measurements of the different resistors (U1(t1)..Un(t1)) is zero. This mean is called the "ensemble mean".

An ergodic signal is a signal for which the ensemble mean and the mean over time are equal. Ergodic signals have to be stationary and can be recognized by the equivalence of ensemble means and means over time.

Counterexample

The German stock index (DAX) is neither ergodic nor stationary. The DAX is a weighted mean of the 30 most important stock prices (it is an ensemble mean). This ensemble mean varies with time, thus it is not stationary. On the other hand the means over time of the individual stocks is not the same as the mean over time of the DAX. Thus the DAX is not an ergodic signal.