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 $X(t)$ 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 mean-ergodic or mean-square ergodic in the first moment 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 autocovariance-ergodic or mean-square ergodic in the second moment 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.

An important example of an ergodic processes is the stationary Gaussian process with continuous spectrum.

Discrete-time random processes

The notion of ergodicity also applies to discrete-time random processes $X[n]$ for integer $n$.

A discrete-time 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 non-ergodic 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 long-term average is ½ for the fair coin and 1 for the two-headed coin. Hence, this random process is not ergodic in mean.

See also

• Ergodic hypothesis
• Ergodicity
• Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
• Loschmidt's paradox
• Poincaré recurrence theorem