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Analytic Properties of Random Processes

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Properties  ::  Continuity  ::  Differentiation  ::  Integration

Integration

For a function $f: T \rightarrow \Rbb$ , we define a Riemann integral as

\begin{displaymath}\lim_{\max\vert t_i - t_{i-1}\vert\rightarrow 0} \sum_{i=1}^n f(\zeta_i)(t_i
- t_{i-1}) = \int_a^b f(t) dt
\end{displaymath}

where $a = t_0 < t_1 < \cdots < t_n = b$ and $z_i \in (t_{i-1},t_i)$ .

Let us define a similar sort of limit for a random process. We will define the limits in the mean-square sense. Then

\begin{displaymath}\int_a^b X_t dt
\end{displaymath}

is the mean-square integral of $X_t$ .

Properties of M.S. integrals $\int_a^b X_t dt$ :

  1. The integral exists if and only if

    \begin{displaymath}\int_a^b \int_a^b R_X(t,s) dtds < \infty.
\end{displaymath}


    \begin{proof}
$\int_a^b X_t dt$ exists if and only if
\begin{displaymath}\beg...
...gral
exists implies that $\int_a^b \int_a^b R_X(t,s) dtds$ exists.
\end{proof}
  2. Assume that $\int_a^b X_tdt$ exists. Then $E[\int_a^b X_t dt]
= \int_a^b E[X_t] dt = \int_a^b \mu_x(t) dt$ .
  3. $E[(\int_a^b X_t dt)^2] = \int_a^b \int_a^b R_X(t,s) dtds$ .
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_4.htm. This work is licensed under a Creative Commons License Creative Commons License