Measuring spatial dispersion: exact results on the variance of random spatial distributions

Measuring the spatial distribution of locations of many entities (trees, atoms, economic activities, etc.), and, more precisely, the deviations from purely random configurations, is a powerful method to unravel their underlying interactions. Several coefficients have been developed in the past to quantify the possible deviations. It is important to quantify the variances of the coefficients for random distributions, to ascertain the statistical significance of an empirical deviation. By lack of a proper analytical expression, the significance is usually obtained by simulating many random configurations by Monte Carlo simulations. In the present paper, we present an exact analytical expression for the variance of several spatial coefficients for random distributions, and we rigorously show that these distributions asymptotically follow a Normal law. These two results eliminate the need for cumbersome Monte Carlo simulations. They also allow to understand qualitatively the main factors that may change the variance: number of sites, spatial inhomogeneity, etc.