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 <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\tilde{\tau}$</EquationSource> </InlineEquation> 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 <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\tau \ge \tilde{\tau}$</EquationSource> </InlineEquation>. Moreover, we show that if the scaling function <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\mathcal{G}(x)$</EquationSource> </InlineEquation> approaches a non-zero constant for small arguments, <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\lim_{x \to 0} \mathcal{G}(x) > 0$</EquationSource> </InlineEquation>, then <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$\tau=\tilde{\tau}$</EquationSource> </InlineEquation>. However, if the scaling function vanishes for small arguments, <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$\lim_{x \to 0} \mathcal{G}(x)=0$</EquationSource> </InlineEquation>, 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. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008