1/f noise in Power spectrum Leads to Long-memory effects

Whilst it is relatively simple to yield power-law (scale-free) relationships in the correlation function (see below), it is of particular interest if the relationship is of the form,

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.. so called '1/f noise', 'flicker noise', or 'pink noise' (see w: link on parent page). The reasoning is as follows:

1. We have the temporal correlation function:

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   and the cosine transformed power spectrum

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   ,

2. Now, suppose

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   and

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   ,

3. Notice that given

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   on the LHS we have

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   for the power spectrum, whilst on the right hand side we have

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   which is a linear function of

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   . Now

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   is the interval of correlation, or in other words the period of correlation. Hence, since we assume

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   , we are actually assuming,

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   ;

4. Combine this with the cosine transform of f in the power spectrum and we have that (LHS ~ RHS):

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That is, if we find that

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(as in flicker noise), then this implies that

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, which causes the assumed relationship for

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to break down and instead be replaced by slow logarithmic decay. Very low-frequency signals have significant impact on inter-temporal events. Hence, the interest in 1/f distributions in the power function.

Ref: p.9 of Jensen

Superposition of Random(Poisson) processes Lead to