Discrete Spacings

Consider a string of n positions, i.e. a discrete string of length n . Units of length k are placed at random on this string in such a way that they do not overlap, and as often as possible, i.e. until all spacings between neighboring units have length less than k . When centered and scaled by n ^−1/2 the resulting numbers of spacings of length 1, 2,…,  k −1 have simultaneously a limiting normal distribution as n →∞. This is proved by the classical method of moments.