On normal approximations to U -statistics

Abstract

Let X1,..., Xn be i.i.d. random observations. Let S{double-struck} = L{double-struck} + T{double-struck} be a U -statistic of order k ≥ 2 where L{double-struck} is a linear statistic having asymptotic normal distribution, and T{double-struck} is a stochastically smaller statistic. We show that the rate of convergence to normality for S{double-struck} can be simply expressed as the rate of convergence to normality for the linear part L{double-struck} plus a correction term, (var T{double-struck}) ln2(var T{double-struck}), under the condition E{double-struck}T{double-struck}2 < ∞. An optimal bound without this log factor is obtained under a lower moment assumption E{double-struck}|T{double-struck}|α < ∞ for α