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Let {F_n} be a normalized sequence of random variables in some fixed Wienerchaos associated with a general Gaussian field, and assume that E[F_n^4] -->E[N^4]=3, where N is a standard Gaussian random variable. Our main result isthe following general bound: there exist two finite constants c,C>0 such that,for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < Cmax(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and hruns over the class of all real functions with a second derivative bounded by1. This shows that the deterministic sequence max(|E[F_n^3]|, E[F_n^4]-3)completely characterizes the rate of convergence (with respect to smoothdistances) in CLTs involving chaotic random variables. These results are usedto determine optimal rates of convergence in the Breuer-Major central limittheorem, with specific emphasis on fractional Gaussian noise.

Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants