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Basic Concepts of Random Processes

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

Basic definitions and concepts

A {\bf random process} (or {\bf stochastic process} on a pro...
...\Omega,\Fc,P)$, where $T$ is an
indexing set of real numbers.
  • If $T$ is a singleton (one element) then $\{X_t, t\in T\}$ is a r.v.
  • If $T = \{t_1,t_2\}$ , then $\{X_t, t\in T\}$ is a bivariate r.v.
  • If $T$ consists of a finite number of elements, then $\{X_t, t\in T\}$ is a random vector.
  • If $T$ is countable, then $\{X_t, t\in T\}$ is a random sequence.
For most applications we think of $t$ as ``time.'' In some cases, $T$ is multidimensional. Then $X_t$ is called a random field.

Three interpretations of a r.p.

  1. A collection of waveforms that occur randomly. That is, it is defined on some probability space. For each $\omega \in \Omega$ there is a corresponding waveform $\{X_t(\omega), t\in T\}$ as a function of $t$ with $\omega$ fixed.

    Think of having a big bag of waveforms. We reach into the bag and pick out a waveform -- a function of $t$ -- at random.

  2. A collection of random variables. In this case, that is, for each fixed $t \in T$ , we have a random variable $X_t$ .
  3. A real-valued function of two variables $X_t: \Omega \times T
\rightarrow \Rbb$ .

A function $\{X_t(\omega), t \in T\}$ assumed by $\{X_t, t ...
... (Also known as a {\bf sample function} or {\bf sample
A realization is just a function. It does not exhibit the randomness.
If $T$ contains a continuum of values (e.g., $T = \Rbb$ or...
...$\{X_t, t \in T\}$ is a {\bf continuous-time random

If $T$ contains only countably many values (e.g. $T = \Zbb$...
...n $\{X_t, t \in T\}$ is a {\bf discrete-time random

Let $n$ be a positive integer and $\{X_t, t \in T\}$ a ran...
...te-dimensional distributions} (f.d.d.s)
of $\{X_t, t \in T\}$.
We will assume this set completely characterizes the statistical distribution of the process.

$T$ is closed under addition if $T_1, T_2 \in T$ implies $T_1 + T_2
\in T$.

Suppose $T$ is closed under addition. The random process $\...
...e for all orders $k$, the r.p. is {\bf (strictly)
Strict stationarity is a fairly strong condition, and we don't necessarily need it always.

Stationarity to order 1 means that $F_{X_t}$ is the same for e...
...X_s}$ depends only on the
{\em difference} between $t$ and $s$.

Copyright 2008, Todd Moon. 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