Asymptotic Behavior of Sum-functions of $$\boldsymbol{m}$$-Tuples of Uniform Spacings

Let n points be taken at random on a circle of unit circumference and clockwise ordered. Uniform spacings are defined as the clockwise arc-lengths between the successive points from this sample. We are interested in the asymptotic behavior of the sums of functions of the $$m$$ -tuples of successive spacings. We derive asymptotic formulas for the expectation and variance of such sums and establish the asymptotic normality of the sums.