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

On the scaling of probability density functions with apparentpower-law exponents less than unity,10.1140/epjb/e2008-00173-2,European Physical Journal

On the scaling of probability density functions with apparentpower-law exponents less than unity  
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.  We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent 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 . Moreover, we show that if the scaling function approaches a non-zero constant for small arguments, 0$" align="middle" border="0"> , then . However, if the scaling function vanishes for small arguments, , 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.
Journal: European Physical Journal B - EUR PHYS J B , vol. 62, no. 3, pp. 331-336, 2008
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