Asymptotic Normality of Sum-Functions of Spacings

Take $n$ points at random on a circle of unit circumference and order them clockwise. Let $S^{(m)}_0,\cdots, S^{(m)}_{n-1}$ be the $m$th order spacings, i.e., the clockwise arc-lengths between every pair of points with $m - 1$ points between. Ordinary spacings correspond to the case $m = 1$. A central limit theorem is proved for $Z_n = \sum^{n-1}_{k=0}h(nS_k,\cdots, nS_{k+m-1})$, where $h$ is a given function. Using this, asymptotic distributions of central order statistics and sums of the logarithms of $m$th order spacings are derived.