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Central Limit Theorems and ‘LogNormal’ distributions

Using CLT, I predicted the distribution of LogZeta in the neighbourhood of any root of Zeta is Rayleigh1 (Section 6). Keating/Snaith have shown the real and imaginary parts of log{ ζ(˝ + iT)} are independent, identical Normal distributions as height on the Critical Line T→∞ and this result has been known for some time.

In the neighbourhood of a zero (and anywhere on or off the Critical Line Re(z)=˝), in the asymptotic limit, the magnitude of LogZeta = R =√(X2 + Y2) can be shown Rayleigh distributed where X and Y are identical, mutually independent Normal distributions each with mean zero N (0, 1); variance is one by normalization. In Sonar, which is my area of expertise, they are said to be ‘LogNormal’ (in Zeta). (Interestingly, Real and Imaginary spectral components have actually been measured in all manner of variety of physical systems that involve thermal noise).

A treatment of the various statistical distributions in sonar and radar is provided by clicking on the following link, which gives a nice derivation of the Ricean distribution.

It is worth mentioning the treatment of spectrum components in Sonar. If X and Y are each the sum of real and imaginary spectral components respectively. Then X and Y are mutually independent in the limit as the number of samples n →∞, even if the individual components themselves have dependency on others. The reason for this is as follows: 

Each of the real and imaginary sums are composed of a series of trigonometric cosines and sines respectively. Inverse trigonometric functions always have 2 values between the range [0, 2p] e.g. if cos(q1) = 0, then q1 = 90° or 270° i.e. 2 possibilities for the sin(q1) namely  ±1. Given known values for each cos(q1) and cos(q2) then there are 4 possible values for {sin(q1) + sin(q2)}. With an infinite series, the total possibilities for the sum of sine terms become endless, so that the X and Y become increasingly uncorrelated and thus independent in the limit

The level spacings di=x i+j -xj are as actually distributed (asymptotically as T→∞) as follows:


  

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