Weak convergence results for the Kakutani interval splitting procedure. (Q1879858)

The paper addresses comparisons between two probability models for random subdivisions of the unit interval: the usual \(U\)-model in which the division points are independent Unif(0,1) random variables and the Kakutani \(K\)-model. The authors study the weak convergence under the \(K\)-model of the empirical processes for both the division points and their spacings. The limit processes are Gaussian. In particular, the obtained results clarify the differences between the \(U\)- and \(K\)-models. The empirical processes for division points converge in law to Brownian bridges, but the standard deviations for the \(K\)-model are approximately half what they are for \(U\)-model. Finally, the covariance function of the limiting empirical processes for the normalized spacings is explicitly derived, thereby characterizing those processes completely.