A Random Interval Filling Problem

Abstract

Consider an initial real interval $\lbrack 0, x\rbrack, x > 0$, and place in it a random subinterval $I(x)$ defined by a pair of random variables $(U_x, V_x)$; the former being the length of $I(x)$ and the latter its lower boundary point. The set $\lbrack 0, x\rbrack - I(x)$ consists of two intervals of lengths $x_1$ and $x_2$, in which there are in turn placed random subintervals $I(x_1)$ and $I(x_2)$ defined by pairs of random variables $(U_{x_1}, V_{x_1})$ and $(U_{x_2}, V_{x_2})$. The process of placing random subintervals in $\lbrack 0, x\rbrack$ is thus continued. Under the assumptions that the subintervals cannot overlap, and that their lengths are uniformly bounded away from zero, the procedure must terminate after a finite number of steps. Let $N(b, x)$ denote the number of subintervals of $\lbrack 0, x\rbrack$ of length at least $b$, in the terminal state. The asymptotic behaviour of the moments of $N(b, x)$ is here studied as $x \rightarrow \infty$. It is shown that under fairly general conditions the mean approaches a linear function of $x$ at the rate $x^{-n}$, for any integer $n > 0$. Under the further condition that $V_x$ is a family of uniform distributions the exact form of the linear relation is determined. In the last section it is indicated how this result can be extended to some more general distributions. A similar but less precise result is proved for the higher moments, the convergence rate $x^{-n}$ not being established for this case.