This is an expository survey of recent developments in the field of rates of convergence and asymptotic expansions in the context of the multi-dimensional central limit theorem. A number of applications are discussed. One of them deals with normal approximations and expansions of distribution functions of a class of statistics which includes functions of sample moments.
"Refinements of the Multidimensional Central Limit Theorem and Applications." Ann. Probab. 5 (1) 1 - 27, February, 1977. https://doi.org/10.1214/aop/1176995887