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finite variance. A tail with a nonconvergent decay rate of 1 < α < 2 indicates non-finite variance in the data such that the usual “normal curve” derived, standard deviation dependent tests of statistical significance are without meaning. α < 1 indicates the data is without a consequential mean and will require the use of interquartile measures to locate the center of the distribution (Adler et al, 1998).
Recalling that the Hurst, Fano and Allan indices are invariant across sample segment size, we remind ourselves that, as is the case in the finite mean and variance, α = 2, Gaussian, any of the other “α tails” also retain their value (“shape”) across all partitions that might be used to sort and sum the observable. This property is called convolutional, α, stability. In passing it should be noted that the last outpost of convergence of a probability density distribution with α = 2 is called “log-normal,” in which the tails along the x axis are “pulled in” by the variable being plotted as its logarithm.

A Hurst exponent of > 0.5 in the data is associated with a Levy exponent of < 2.0, and both would be indicative of a process in which the characteristic style of change, rather than decay with some finite correlation length, would persist across all time. Using a bursting neuron as a generic example, a short interspike interval would, on the average, be followed by another short one and a long one by another long one, and this behavior, unlike our fair coin flipping sequence of observables, would not become uncorrelated with itself even over infinite time. Another way to represent this infinite, innumerably lengthed, correlation property is via its implicate frequency (inverse wavelength) content by computing its best fit assortment (along with their densities) of a range of short to long sine waves forming the Fourier transformation of the correlation function. The condition of correlated fluctuations across many measured temporal scales yields yet another power law slope when graphed as the logarithm of its range of frequencies, f, plotted along the x axis, versus their corresponding amplitudes squared, powers, plotted along the y axis. Naming this spectral power law exponent β, the system’s characteristic scaling law is usually expressed as 1/f^β (Fedor, 1988; Hughes, 1995; Shlesinger, 1996;

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