Ergodic process

In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birds-eye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate.
Specific definitions.
One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process formula_1 has constant mean
and autocovariance 
that depends only on the lag formula_4 and not on time formula_5. The properties formula_6 and formula_7
are ensemble averages not time averages.
The process formula_1 is said to be mean-ergodic or mean-square ergodic in the first moment
if the time average estimate
converges in squared mean to the ensemble average formula_6 as formula_11.
Likewise,
the process is said to be autocovariance-ergodic or d moment 
if the time average estimate
converges in squared mean to the ensemble average formula_7, as formula_11.
A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.
Discrete-time random processes.
The notion of ergodicity also applies to discrete-time random processes
formula_15 for integer formula_16.
A discrete-time random process formula_15 is ergodic in mean if
converges in squared mean
to the ensemble average formula_19,
as formula_20.
Examples.
Ergodicity means the ensemble average equals the time average. Following are examples to illustrate this principle. 
Call centre.
Each operator in a call centre spends time alternately speaking and listening on the telephone, as well as taking breaks between calls. Each break and each call are of different length, as are the durations of each 'burst' of speaking and listening, and indeed so is the rapidity of speech at any given moment, which could each be modelled as a random process. 
Electronics.
Each resistor has an associated thermal noise that depends on the temperature. Take "N" resistors ("N" should be very large) and plot the voltage across those resistors for a long period. For each resistor you will have a waveform. Calculate the average value of that waveform; this gives you the time average. There are "N" waveforms as there are "N" resistors. These "N" plots are known as an ensemble. Now take a particular instant of time in all those plots and find the average value of the voltage. That gives you the ensemble average for each plot. If ensemble average and time average are the same then it is ergodic.