High order joint moments calculation

Wu Xiaobin

Recently I have been working on classical Gaussian electrical field and I come through this joint
moments calculation.
Suppose we got the joint density function as:
[itex]
p(s_i,s_j)=\frac{1}{2\pi d}\exp{[\frac{1}{2d}(<s_i>s_i+<s_j>s_j)]}K_0(\frac{1}{2d}\sqrt{<I>^2-s_k^2}\sqrt{s_i^2+s_j^2})
[/itex]
[itex]<I>,<s_i>,<s_j>,<s_k>[/itex] are the known mean of [itex]I,s_i,s_j,s_k[/itex].
the high order moment of [itex]<s_i^n>[/itex] can be calculated as:
[itex]
<s_i^n>=\frac{n!}{2^{n+1}\sqrt{<I>^2-<s_j>^2-<s_k>^2}}[(\sqrt{<I>^2-<s_j>^2-<s_k>^2}+<s_i>)^{n+1}+(-1)^n(\sqrt{<I>^2-<s_j>^2-<s_k>^2}-<s_i>)^{n+1}]
[/itex]
with the above result, The paper I referred to give the following conclusion which I can't catch up with:
[itex]
<s_i^ns_j^m>=(\frac{2d}{<s_j>})^m[\frac{\partial^m<s_i^n(x,s_j)>}{\partial x^m}]_{x=1}
[/itex]
where
[itex]
<s_i^n(x,s_j)>
[/itex]
is given by multiplying [itex]<s_j>^2[/itex], in the square brackets in [itex] <s_i^n>[/itex]
Can anyone tell me why the author calculates the joint moments in that way.

Look forward to your reply
Sincerely yours
Jacky Wu

Attached Files

Statistics of Stokes variables for correlated Gaussian Fields.pdf (255.0 KB, 0 views)