Extreme values of random or chaotic discretization steps and connected networks

By sorting independent random variables and considering the difference between two consecutive order statistics, we get random variables, called steps or spacings, that are neither independent nor identically distributed. We characterize the probability distribution of the maximum value of these steps, in three ways: i/with an exact formula; ii/with a simple and finite approximation whose error tends to be controlled; iii/with asymptotic behavior when the number of random variables drawn (and therefore the number of steps) tends towards infinity. The whole approach can be applied to chaotic dynamical systems by replacing the distribution of random variables by the invariant measure of the attractor when it is set. The interest of such results is twofold. In practice, for example in the telecommunications domain, one can find a lower bound for the number of antennas needed in an ad hoc network to cover an area. In theory, our results take place inside the extreme value theory extended to random variables that are neither independent nor identically distributed.