Asymptotics of moments in the central limit theorem in Banach spaces (Q762824)

Let B denote a separable Banach space with norm \(| \cdot |\). Let \(X\in B\) be a random element with mean zero, \(S_ n=(X_ 1+...+X_ n)/\sqrt{n}\) be the normalized sum of n independent copies of X, \(Y\in B\) be the Gaussian element with mean zero and covariance equal to that of X. Denote \(\Delta_ n(q)=E| S_ n|^ q-E| Y|^ q\). If the norm of the Banach space B is sufficiently smooth in the sense of Fréchet in the region \(B\setminus \{0\}\), then \(\Delta_ n(q)=o(n^{- (p-2)/2})\) if \(E| X|^ p<\infty\), \(q\leq p\), \(2<p<4\); \(\Delta_ n(q)=o(n^{-1})\) if \(E| X|^ p<\infty\), \(q\leq p\), \(p\geq 4\). In case of the Hilbert space the mentioned estimates of \(\Delta_ n(q)\) hold under the moment condition \(E| X| <\infty\) only.