Personal tools
  •  
You are here: Home Electrical and Computer Engineering Stochastic Processes Analytic Properties of Random Processes

Analytic Properties of Random Processes

Document Actions
  • Content View
  • Bookmarks
  • CourseFeed

Properties  ::  Continuity  ::  Differentiation  ::  Integration

Continuity

Let $T = (a,b)$ . Recall that a function $f: T \rightarrow \Rbb$ is continuous at $t_0$ if $\lim_{t \rightarrow t_0} f(t) =
f(t_0)$ . $f$ is continuous if it is continuous for every $t_0 \in
(a,b)$ .


\begin{definition}
For a random process $\{X_t, t \in T\}$, we say $X_t$ is {\...
... \Omega:
X_t(\omega), t \in T \text{ is continuous}\}) = 1.$\
\end{definition}
Continuity w.p. 1 is usually quite strong, in fact stronger than is typically needed for analysis.

We say that $\{X_t, t\in T\}$ is continuous in probability at $t_0$ if $X_t \rightarrow X_{t_0}$ (i.p.) as $t \rightarrow t_0$ . That is,

\begin{displaymath}\lim_{t \rightarrow t_0} P(\vert X_t - X_{t_0}\vert > \epsilon) = 0 \text{
for all } \epsilon > 0.
\end{displaymath}

We say that $\{X_t, t\in T\}$ is mean-square continuous at $t_0$ if $X_t \rightarrow X_{t_0}$ (m.s.) as $t \rightarrow t_0$ . That is,

\begin{displaymath}\lim_{t \rightarrow t_0} E[\vert X_t - X_{t_0}\vert^2] = 0.
\end{displaymath}

An important property: A second-order random process is mean-square continuous at $t=t_0$ if and only if $R_X(t,s)$ is continuous at $t=s=t_0$ , that is

\begin{displaymath}\lim_{t \rightarrow t_0,s\rightarrow t_0} R_X(t,s) = R_X(t_0,t_0).
\end{displaymath}


\begin{proof}
(If) Suppose $R_X(t,s)$ is continuous at $t=s=t_0$. Then
\begin{...
... $\rightarrow 0$. Thus $R_{X}(t,s)$ is continuous at
$t = s = t_0$.
\end{proof}

\begin{example}
(Homogeneous Poisson counting process). $R_X(t,s) = \lambda
\m...
...es of jumps: any realization is
discontinuous with probability 1.
\end{example}

\begin{example}
Let $X_t$ be a Gaussian random process with
\begin{displaymath...
...ay about it later.) It models random diffusion or Brownian motion.
\end{example}

For a W.S.S. process, we have the following: A W.S.S. r.p. $X_t$ is mean-square continuous if and only if $R_X(\tau)$ is continuous at $\tau = 0$ . This follows since $R_X(t,s) = R_X(t-s)$ , and

\begin{displaymath}\lim_{(t,s) \rightarrow (t_0,t_0)} R_X(t,s) = \lim_{\tau
\rightarrow 0} R_X(\tau).
\end{displaymath}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Analytic Properties of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture7_2.htm. This work is licensed under a Creative Commons License Creative Commons License