Lower estimation of the remainder term in the CLT for a sum of the functions of k-spacings

The main result is lower estimation of the remainder term in the central limit theorem for a sum of the functions of disjoint uniform k-spacings. From this result, the Lindeberg's type condition of asymptotical normality and Berry-Esseen's type bound is derived. Note that here k=k(n) may tend to infinity if sample size n→∞.