On normal approximations to U -statistics

Let X-1, ..., X-n be i.i.d. random observations. Let S = L + T be a U-statistic of order k >= 2 where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate of convergence to normality for the linear part L plus a correction term, (varT) ln(2) (varT), under the condition ET2 < infinity. An optimal bound without this log factor is obtained under a lower moment assumption E vertical bar T vertical bar(alpha) < infinity for alpha < 2. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature