Accessing power-law statistics under experimental constraints

Power-law distributions appear in a large variety of situations and influence our understanding of various physical phenomena. Their identification and characterization are notoriously difficult because of the large fluctuations inherent to empirical data and also because of the unknown range over which the power-law behavior holds. Furthermore, the data on which one is trying to detect power laws are affected by technical constraints and experimental limitations. Here, we show how a power-law distribution is modified by two fundamental limitations: the spatiotemporal resolution and the time window. We consider a time series of events or states and investigate the interevent time probability density function (PDF) or the PDF of the duration of a state. We present in detail how each limitation affects the PDF and derive mathematical expressions that relate the observed distribution to the true one: the resolution globally affects the shape of PDF while preserving the asymptotic exponent and the time window introduces a nonexponential cutoff. We demonstrate that, instead of looking for a simple power law in experimental data, one should fit the data with the modified PDF that we derived for given experimental constraints. We apply our theory to data from an experimental study of the transport of mRNA-protein complexes along dendrites. The presented mathematical theory widens our understanding of the identification and characterization of power-law distributions in experimental data and can be used in a broad spectrum of science fields.