P(x) \propto e^{-\lambda x}

has an excess kurtosis of 6 for arbitrary decay constants.

Different values of the kurtosis can be obtained for distributions that are superpositions of exponential distributions with different decay constants.

As a test, the distribution

P(x) \;\propto\; 10 e^{- 10 x}\; + \; e^{-x}

has been sampled numerically. The resulting set of random numbers has then been analyzed statistically:

The numerical kurtosis of the superposition is 17.

This could be relevant for the "spider web model of cytoskeletal fluctuations": An elementary remodeling step in a remote fiber causes a smaller shift of the bead, i.e. the corresponding probability distribution of shifts has a steeper decay. The total shift distribution is a superposition of the different fiber contributions.

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