[AD 4] Positive kurtosis comes from shaped noise

We have shown before that a narrowed phase distribution yields a positive kurtosis at small lag times. It remains to be shown if the noise or the PL part is responsible for this behaviour.

Following graph: Kurtosis vs lag-time for wide phase distribution, when only noise or only PL is present:

Pure noise has no kurtosis and pure PL has a kurtosis decaying from zero to negative values.

Following graph: The same for narrowed phase distribution:


So it is clearly the noise part that creates positive kurtosis. The noise is shaped by the narrowed phase distribution in a way that makes the step width (= average velocity * time step) distribution non-Gaussian (compatible with the VAC).

This conclusion is reasonable: In the range of small lag times, the noise part anyway dominates over the PL part. Therefore, the PL-VAC cannot be responsible for the positive kurtosis.