Power Law - Power-law Probability Distributions

Power-law Probability Distributions

In the most general sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form

where, and is a slowly varying function, which is any function that satisfies with constant. This property of follows directly from the requirement that be asymptotically scale invariant; thus, the form of only controls the shape and finite extent of the lower tail. For instance, if is the constant function, then we have a power law that holds for all values of . In many cases, it is convenient to assume a lower bound from which the law holds. Combining these two cases, and where is a continuous variable, the power law has the form

where the pre-factor to is the normalizing constant. We can now consider several properties of this distribution. For instance, its moments are given by

which is only well defined for . That is, all moments diverge: when, the average and all higher-order moments are infinite; when, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge - as more data is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails.

Another kind of power-law distribution, which does not satisfy the general form above, is the power law with an exponential cutoff

In this distribution, the exponential decay term eventually overwhelms the power-law behavior at very large values of . This distribution does not scale and is thus not asymptotically a power law; however, it does approximately scale over a finite region before the cutoff. (Note that the pure form above is a subset of this family, with .) This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. For instance, although the Gutenberg–Richter law is commonly cited as an example of a power-law distribution, the distribution of earthquake magnitudes cannot scale as a power law in the limit because there is a finite amount of energy in the Earth's crust and thus there must be some maximum size to an earthquake. As the scaling behavior approaches this size, it must taper off.