On the scaling of probability density functions with apparent power-law exponents less than unity

We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent ($) over tilde of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent center dot equal to 1, provided that the fraction of events in the universal scaling part of the probability density function is non-vanishing in the thermodynamic limit. We find the general result that center dot >= 1 and center dot >= (center dot) over tilde. Moreover, we show that if the scaling function G(center dot) approaches a non-zero constant for small arguments, lim(x -> 0) G(center dot) center dot 0, then center dot = (center dot) over tilde. However, if the scaling function vanishes for small arguments, lim(x -> 0) G(center dot) = 0, then center dot = 1, again assuming a non-vanishing fraction of universal events. Finally, we apply the formalism developed to examples from the literature, including some where misunderstandings of the theory of scaling have led to erroneous conclusions.