Relating exponents of truncated power laws for self-similar signals

Stochastic simulations are conducted to evaluate relations between statistics for processes governed by a power law probability distribution with exponent $\alpha$. The power law exponent is determined for statistics of simulated signals, namely the box-counting fractal dimension $D$, energy spectrum exponent $\beta$, and an intermittency exponent $\mu$. For a binary signal with no variability in amplitude, the parameters are related linearly as $D = 2 - \beta = 1 - \mu$. The relations are unchanged if the sampled power law distribution is truncated to a finite range of values, e.g. for a distribution exhibiting a cutoff. However, truncating the distribution yields statistics that are not truly scale-invariant, and distorts the connection between the statistic exponents and $\alpha$. The behavior is due to the survival function, or the complementary cumulative distribution, which for a finite-sized power law is only approximately self-similar and has an effective exponent differing from $\alpha$. An expression for the effective exponent is presented. The results are discussed in the context of turbulent flows, but are generally applicable to any statistically self-similar signal.