Diversity of Poissonian populations

Populations represented by collections of points scattered randomly on the real line are ubiquitous in science and engineering. The statistical modeling of such populations leads naturally to Poissonian populations—Poisson processes on the real line with a distinguished maximal point. Poissonian populations are infinite objects underlying key issues in statistical physics, probability theory, and random fractals. Due to their infiniteness, measuring the diversity of Poissonian populations depends on the lower-bound cut-off applied. This research characterizes the classes of Poissonian populations whose diversities are invariant with respect to the cut-off level applied and establishes an elemental connection between these classes and extreme-value theory. The measures of diversity considered are variance and dispersion, Simpson’s index and inverse participation ratio, Shannon’s entropy and Rényi’s entropy, and Gini’s index.