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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


\begin{definition} A {\bf random process} (or {\bf stochastic process} on a pro... ...\Omega,\Fc,P)$, where $T$ is an indexing set of real numbers. \end{definition}
  • 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 $X_t: \Omega \times T \rightarrow \Rbb$.

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

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

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


\begin{definition} $T$ is closed under addition if $T_1, T_2 \in T$ implies $T_1 + T_2 \in T$. \end{definition}

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


\begin{example} 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 $s$. \end{example}

Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture6_1.htm. This work is licensed under a Creative Commons License. Creative Commons License
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