The thresholding of time series of activity or intensity is frequently used to define and differentiate
events. This is either implicit, for example due to resolution limits, or explicit, in order to filter certain
small scale physics from the supposed true asymptotic events. Thresholding the birth–death process,
however, introduces a scaling region into the event size distribution, which is characterized by an
exponent that is unrelated to the actual asymptote and is rather an artefact of thresholding. As a result,
numerical fits of simulation data produce a range of exponents, with the true asymptote visible only in
the tail of the distribution. This tail is increasingly difficult to sample as the threshold is increased. In
the present case, the exponents and the spurious nature of the scaling region can be determined
analytically, thus demonstrating the way in which thresholding conceals the true asymptote. The
analysis also suggests a procedure for detecting the influence of the threshold by means of a data
collapse involving the threshold-imposed scale.