A Proof of Kakutani's Conjecture on Random Subdivision of Longest Intervals

Abstract

Choose a point at random, i.e., according to the uniform distribution, in the interval (0, 1). Next, choose a second point at random in the largest of the two subintervals into which (0, 1) is divided by the first point. Continue in this way, at the $n$th step choosing a point at random in the largest of the $n$ subintervals into which the first $(n - 1)$ points subdivide (0, 1). Let $F_n$ be the empirical distribution function of the first $n$ points chosen. Kakutani conjectured that with probability 1, $F_n$ converges uniformly to the uniform distribution function on (0, 1) as $n$ tends to infinity. It is shown in this note that this conjecture is correct.