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Basic Concepts of Random Processes

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Definitions  ::  Ergodicity  ::  Autocorrelations  ::  Sinusoidal  ::  Poisson  ::  Gaussian  ::  Properties  ::  Spectra  ::  Cases  ::  Random  ::  Independent


Assume throughout that $\{X_t\}$ is stationary.

Loosely speaking, a random process $\{X_t\}$ is ergodic if time averages are equal to ensemble averages. That is, averages over $\omega$ -- expectations -- are the same as averages over $t$ . That is, ensemble averages are the same as sample averages.

Here is an example: Suppose $\{X_n\}$ is an i.i.d. sequence. The ensemble mean is $\mu = \int X(\omega) P(d\omega)$ . The sample mean is

\begin{displaymath}\frac{1}{n} \sum_{k=1}^n X_k(\omega).

By the S.L.L.N. we have

\begin{displaymath}\frac{1}{n} \sum_{k=1}^n X_k(\omega) \rightarrow \mu.

This is an example of an ergodic property.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: This work is licensed under a Creative Commons License Creative Commons License