We now come to the statistics of . For example, we already
. How about
? Quite easy to check that it is zero because
when we let the 's each vary independently over the full circle to . This is true whether are distinct or not. But coming to even powers like , something interesting happens. When we integrate a product like over all the four 's we can get non-zero answers, provided the 's occur in pairs, i.e., if and , then we encounter which has a non-zero average. (We saw a particular case of this when we calculated and only type terms survived).
Because of the random and independent phases of the large number
of different frequencies, we can now state the ``pairing theorem''.
As discussed in Section 1.8, this pairing theorem proves that the statistics is gaussian. (A careful treatment shows that only the terms are equal on the two sides- we have not quite got the terms right, but there are many more (of the order of times more) of the former type and they dominate as and the numbers of sines and cosines we are adding is very large). This result -- that the sum of a large number of small, finite variance, independent terms has a gaussian distribution -- is a particular case of the ``central limit theorem''. We only need the particular case where these terms are cosines with random phases.