In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties (such as its mean and variance) can be deduced from a single, sufficiently long sample (realization) of the process. The reasoning behind this is that any sample from the process must represent the average statistical properties of the entire process, so that regardless what sample you choose, it represents the whole process, and not just that section of the process. A process that changes erratically at an inconsistent rate is not said to be ergodic.^{[1]}
Contents

Specific definitions 1

Discretetime random processes 2

Example of a nonergodic random process 3

See also 4

Notes 5

References 6
Specific definitions
One can discuss the ergodicity of various statistics of a stochastic process. For example, a widesense stationary process X(t) has constant mean

\mu_X= E[X(t)],
and autocovariance

r_X(\tau) = E[(X(t)\mu_X) (X(t+\tau)\mu_X)],
that depends only on the lag \tau and not on time t. The properties \mu_X and r_X(\tau) are ensemble averages not time averages.
The process X(t) is said to be meanergodic^{[2]} or meansquare ergodic in the first moment^{[3]} if the time average estimate

\hat{\mu}_X = \frac{1}{T} \int_{0}^{T} X(t) \, dt
converges in squared mean to the ensemble average \mu_X as T \rightarrow \infty.
Likewise, the process is said to be autocovarianceergodic or meansquare ergodic in the second moment^{[3]} if the time average estimate

\hat{r}_X(\tau) = \frac{1}{T} \int_{0}^{T} [X(t+\tau)\mu_X] [X(t)\mu_x] \, dt
converges in squared mean to the ensemble average r_X(\tau), as T \rightarrow \infty. A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.^{[3]}
An important example of an ergodic processes is the stationary Gaussian process with continuous spectrum.
Discretetime random processes
The notion of ergodicity also applies to discretetime random processes X[n] for integer n.
A discretetime random process X[n] is ergodic in mean if

\hat{\mu}_X = \frac{1}{N} \sum_{n=1}^{N} X[n]
converges in squared mean to the ensemble average E[X], as N \rightarrow \infty.
Example of a nonergodic random process
Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins, and then perform a sequence of independent tosses of our selected coin. Let X[n] denote the outcome of the nth toss, with 1 for heads and 0 for tails. Then the ensemble average is ½ · ½ + ½ · 1 = ¾; yet the longterm average is ½ for the fair coin and 1 for the twoheaded coin. Hence, this random process is not ergodic in mean.
See also
Notes

^ Originally due to L. Boltzmann. See part 2 of ('Ergoden' on p.89 in the 1923 reprint.) It was used to prove equipartition of energy in the kinetic theory of gases

^ Papoulis, p.428

^ ^{a} ^{b} ^{c} Porat, p.14
References
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