  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 sum of functions of the m-tuples of successive
spacings under the assumption that m can grow together with n. Asymptotic
formulas for the expectation and variance of the sum and its asymptotic
normality are established.
