  We consider a random interval splitting process, in which the splitting rule
depends on the empirical distribution of interval lengths. We show that this
empirical distribution converges to a limit almost surely as the number of
intervals goes to infinity. We give a characterization of this limit as a
solution of an ODE and use this to derive precise tail estimates. The
convergence is established by showing that the size-biased empirical
distribution evolves in the limit according to a certain deterministic
evolution equation. Although this equation involves a non-local, non-linear
operator, it can be studied thanks to a carefully chosen norm with respect to
which this operator is contractive.
  In finite-dimensional settings, convergence results like this usually go
under the name of stochastic approximation and can be approached by a general
method of Kushner and Clark. An important technical contribution of this
article is the extension of this method to an infinite-dimensional setting.
