Length laws for random subdivision of longest intervals (Q1082707)

The author proves a theorem which strengthens the conjecture that Kakutani-splitting of an interval is the limit of \(\alpha\)-splitting as \(\alpha\to \infty.\) The following stochastic model is called Kakutani-splitting: At time \(t=0\) there is one interval, [0,1]. If at time n there are \(N_ n\) subintervals of [0,1], then at time \(n+1\) all longest subintervals, say of length L, split simultaneously and independently into a left interval of length LV and a right interval of length L(1-V), where V has a distribution function on (0,1). In the case of \(\alpha\)-splitting, a real number \(\alpha\) determines which subinterval is split. At time n, there are \(n+1\) subintervals of lengths \(L_ 1,L_ 2,...,L_{n+1}\). One subinterval, to split, is chosen according to the probability density that assigns mass proportional to \(L_ i^{\alpha}\) to the i-th interval.