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

Abstract.We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent $\tilde{\tau}$ of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent τ 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 τ≥1 and $\tau \ge \tilde{\tau}$. Moreover, we show that if the scaling function $\mathcal{G}(x)$ approaches a non-zero constant for small arguments, $\lim_{x \to 0} \mathcal{G}(x) > 0$, then $\tau = \tilde{\tau}$. However, if the scaling function vanishes for small arguments, $\lim_{x \to 0} \mathcal{G}(x) = 0$, then τ= 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.