Ergodic process

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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.

Specific definitions

One can discuss the ergodicity of various properties of a stochastic process. For example, a wide-sense stationary process x(t) has mean m_{x}(t)=E[x(t)] and autocovariance r_{x}(\tau )=E[(x(t)-m_{x}(t))(x(t+\tau )-m_{x}(t+\tau ))] which do not change with time. One way to estimate the mean is to perform a time average:

{\hat {m}}_{x}(t)_{{T}}={\frac {1}{2T}}\int _{{-T}}^{{T}}x(t)\,dt.

If {\hat {m}}_{x}(t)_{{T}} converges in squared mean to m_{x}(t) as T\rightarrow \infty , then the process x(t) is said to be mean-ergodic[1] or mean-square ergodic in the first moment.[2]

Likewise, one can estimate the autocovariance r_{x}(\tau ) by performing a time average:

{\hat {r}}_{x}(\tau )={\frac {1}{2T}}\int _{{-T}}^{{T}}[x(t+\tau )-m_{x}(t+\tau )][x(t)-m_{x}(t)]\,dt.

If this expression converges in squared mean to the true autocovariance r_{x}(\tau )=E[(x(t+\tau )-m_{x}(t+\tau ))(x(t)-m_{x}(t))], then the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment.[2]

A process which is ergodic in the first and second moments is sometimes called ergodic in the wide sense.[2]

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

See also

Notes

  1. ↑ Papoulis, p.428
  2. ↑ 2.0 2.1 2.2 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. 


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