Definition
In a power-law model, the probability of exceeding a large threshold is proportional to a negative power of that threshold. The lower boundary of applicability and the parameter must be stated. A density, an exceedance probability and the relationship between size and rank have different definitions; their coefficients cannot be transferred mechanically from one to another.[1]
Educational scale check
If, in a specially specified model, the probability of exceeding x is proportional to the inverse square of x, doubling the threshold reduces the probability by a factor of four. This is a property of the formula, not an established law of all financial series.
Limitation
A nearly straight line on a log-scale chart does not select a model by itself. The original observations, threshold, missing values, parameter uncertainty and alternative distributions matter. A small set of the largest companies may poorly represent the tail of the whole population.
A cross-sectional size distribution does not determine the future growth of one company, still less the return on its shares. The distinctions are discussed in [power laws and scaling](/guides/power-laws-scaling/).
Primary material
[1] Gabaix, Power Laws in Economics and Finance; overview of distributions of size and other economic quantities.
[Power Laws in Economics and Finance](https://www.nber.org/papers/w14299)