What is an ergodic time series?

What is an ergodic time series?

In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average.

Why do we need Ergodicity?

The idea behind ergodicity is that, while collecting more and more observations, we keep learning something new about the process. In other words, if I pick two random variables of the process which are sufficiently ‘far apart’, their distributions should be independent among each others.

Is a random walk ergodic?

Theorem 1 A random walk on a graph G is ergodic if and only if G is connected and not bipartite. The necessity is straightforward. G must be connected since otherwise two initial distributions which have all mass in two different components respectively will not converge to the same limiting distribution.

What does the ergodic theorem mean for a process?

An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average. In practice this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process…

What does it mean to have a stationary ergodic process?

Stationary ergodic process. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.

Is the sampled distribution a reflection of an ergodic process?

Although the measured process may be stationary in the long term, it is not appropriate to consider the sampled distribution to be the reflection of a single (ergodic) process: The ensemble average is meaningless. Also see ergodic theory and ergodic process .

Which is a function of a stationary time series?

against jis called the autocorrelation function (ACF). Intuitively, a stationary time series is defined by its mean, variance and ACF. A useful result is that any function of a stationary time series is also a stationary time series. So if {yt} is stationary then {zt} = {g(yt)} is stationary for any function g(·).