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.^[1]

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^[2] or mean-square 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 autocovariance-ergodic or mean-square 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.

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

Notes

1. Originally due to L. Boltzmann. See part 2 of Vorlesungen über Gastheorie. Leipzig: J. A. Barth. 1898. OCLC 01712811. ('Ergoden' on p.89 in the 1923 reprint.) It was used to prove equipartition of energy in the kinetic theory of gases
2. Papoulis, p.428
3. Porat, p.14

References

• Porat, B. (1994). Digital Processing of Random Signals: Theory & Methods. Prentice Hall. p. 14. ISBN 0-13-063751-3.
• Papoulis, Athanasios (1991). Probability, random variables, and stochastic processes. New York: McGraw-Hill. pp. 427–442. ISBN 0-07-048477-5.