Moment Theorem
Theorem:
For a random variable
,
![]() |
(D.47) |
where
![$ \Phi(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img2853.png)
![$ p(x)$](http://www.dsprelated.com/josimages_new/sasp2/img2632.png)
![$ x$](http://www.dsprelated.com/josimages_new/sasp2/img38.png)
![]() |
(D.48) |
(Note that
![$ \Phi(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img2853.png)
![$ p(x)$](http://www.dsprelated.com/josimages_new/sasp2/img2632.png)
Proof: [201, p. 157]
Let
denote the
th moment of
, i.e.,
![]() |
(D.49) |
Then
![\begin{eqnarray*}
\Phi(\omega) &=& \int_{-\infty}^\infty p(x)e^{j\omega x} dx \\
&=& \int_{-\infty}^\infty p(x) \left(1 + j\omega x + \cdots + \frac{(j\omega)^n}{n!}+\cdots\right)dx\\
&=& 1 + j\omega m_1 + \frac{(j\omega)^2}{2} m_2 + \cdots + \frac{(j\omega)^n}{n!}m_n+\cdots
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2857.png)
where the term-by-term integration is valid when all moments
are
finite.
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