A result of C. C. Heyde, where it is shown that certain rates of convergence (expressed in terms of series conditions) in the CLT for i.i.d. random variables hold true if and only if corresponding moment conditions are fulfilled, is generalized to $k$-dimensional i.i.d. random vectors.
R. Michel. "Generalization and Application of a Result of C. C. Heyde." Ann. Probab. 10 (4) 1066 - 1068, November, 1982. https://doi.org/10.1214/aop/1176993729