Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry–Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics. Applications to U-statistics, multisample U-statistics, L-statistics, random sums and functions of nonlinear statistics are discussed.
Louis H.Y. Chen. Qi-Man Shao. "Normal approximation for nonlinear statistics using a concentration inequality approach." Bernoulli 13 (2) 581 - 599, May 2007. https://doi.org/10.3150/07-BEJ5164