Ergodicity

The term ergodicity is vague in itself and possess different meanings in different fields. The word is derived from the Greek word  'ergon' (work) and 'odos' (path), and  was coined by Austrian physicist Ludwig Boltzmann.  There is a separate course in Sigma algebra to deal with ergodicity in detail. In plain words, ergodicity is a term that describes the system which possess same behavior as averaged over time and space.

The main purpose of this article is to make the readers acquainted with the term ergodic and ergodicity, statistically and literally. 

Let us consider an example to understand the concept of ergodicity. This example is given in http://news.softpedia.com/news/What-is-ergodicity-15686.shtml . I  am mentioning the same here because it explain the concept of ergodicity in simple words. 

Suppose we are concerned with determining the most visited parks in the city. One possible way to do so is to follow a single person for a month and see which park does he visit most in that period of time and the other way of figuring out is to take a momentary snapshot of all the parks at a time and see which parks got most of the people at that particular instant of time.  Here, we have got two analyses, one is the statistical data of an individual for a certain period of time and the other is the statistical data of an entire ensemble of people at a particular moment in time. 

So, we see that the situation one is for only a single person,which may not be valid for a large number of people and the statistics depend only on the single individual to whom we considered. While in the situation two it may not be valid for a longer period of time. We are considering only a short span and in the long run, the result may differ. So it is not sure that the statistical results we obtain from both the observations should be the same or alike.

We say that an ensemble is ergodic if both the situations we considered here gives the same statistical results i.e  when the temporal and spatial statistics of an event or a number of samples are same then the event or samples are called ergodic. In simple words, ergodic refers to the dynamical system that has the same behavior averaged over time and averaged over space.

The initial concept in ergodicity is the Poincaré recurrence theorem in statistics which states that certain systems after a sufficiently long time, return to a state very close to the initial state. It means after the long time, the time average exist almost everywhere in the space and is almost the same as the space average.

In signal processing the stochastic process is said to be ergodic if its statistical properties can be deduced from a single long sample or realization of the process (Source wikipedia). If we would like to determine the mean of a stationary stochastic process, we observe a large number of samples and use their ensemble average. However if we have access to only their single sample over a certain period of time, can this time average be used as an estimate for the sample mean. For the ergodic process, the time average should be the estimate of the sample mean. 

 A random process X(t) is said to be ergodic in the mean, i.e., first-order ergodic if the mean of sample average asymptotically approaches the ensemble mean:

 i.e.  lim T tends to infinity E(µxTau) =µx(t)

and lim T tends to infinity var(µxTau) =0