Fractal Scaling of Population Counts Over Time Spans
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Attributes which are infrequently expressed in a population can require weeks or months of counting to reach statistical significance. But replacement in a stable population increases long-term counts to a degree determined by the probability distribution of lifetimes. If the lifetimes are in a Pareto distribution with shape factor 1-r between 0 and 1, then the expected counts for a stable population are proportional to time raised to the r power. Thus r is the fractal dimension of counts versus time for this population. Furthermore, the counts from a series of consecutive measurement intervals can be combined using the L^p-norm where p=1/r to approximate the population count over the combined time span. Data from digital advertising support these assertions and find that the largest reachable fraction of an audience over a long time span is about 1-r.
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