The central limit theorem for sum-functions of m-tuples of spacings

Abstract: 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.