Probabilistic analysis of maximal gap and total accumulated length in interval division

Interval division has been investigated from the point of view of stopping rules. We pay attention here to the quality of the partition. We look at the length of the maximal gap and a certain type of cumulative weights. For the distribution function of the length of the maximal gap we obtain a functional equation, and show how to solve it in sections. The sectional solutions are used to provide successively improved approximations of the average maximal gap. We show that a certain type of cumulative weights asymptotically, when suitably scaled, follows Dickman's infinitely divisible distribution.