Random packing of an interval. II
David Mannion · Advances in Applied Probability · 1979
We showed in [2] that if an object of initial size x ( x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z ( x, B ) denotes the number of such cells whose sizes are points in the set B , then there exists c , (0 < ≦ 1), such that Z ( x, B ) x − c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y 1 , y 2 , ···, y k and if for each k x - y 1 - y 2 - ··· - y k (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the n th cumulant, Ψ n ( x, B ), of Z ( x, B ) satisfies Ψ n ( x, B ) x - c → κ n ( B ), as x → ∞, for some κ n ( B ). Thus, writing N = x c , Z ( x, B ) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.