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The characteristic function of a zero-mean Gaussian is
![$\displaystyle \Phi(\omega) = e^{-\frac{1}{2}\sigma^2\omega^2}$](img2853.png) |
(D.53) |
Since a zero-mean Gaussian
is an even function of
, (i.e.,
), all odd-order moments
are zero. By the moment
theorem, the even-order moments are
![$\displaystyle m_i = \left.(-1)^{\frac{n}{2}}\frac{d^n}{d\omega^n}\Phi(\omega)\right\vert _{\omega=0}$](img2855.png) |
(D.54) |
In particular,
Since
and
, we see
,
, as expected.
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