On the rate of convergence in the central limit theorem in Banach spaces (Q1080255)

Let B be a separable Banach space with norm \(\| \cdot \|\) and \(X_ i\), \(i\geq 1\) be i.i.d. B-valued random elements with \(EX_ 1=0\), \(E\| X_ 1\|^{3+\gamma}<\infty\). Suppose that \(X_ 1\) satisfies the central limit theorem. Let \(S_ n=n^{-1/2}\sum^{n}_{i=1}X_ i\) and \(\eta\) be a Gaussian random element with mean zero and the same covariance as \(X_ 1\). Let \(F: B\to {\mathbb{R}}\) be a measurable functional. In the paper the estimate \[ \sup_{x}| P\{F(S_ n)<x\}- P\{F(\eta)<x\}| =O(n^{-1/2}) \] is proved, assuming that F satisfies some differentiability condition, F and \(X_ 1\) satisfy a so- called ''variance'' condition, which in turn guarantees the existence of bounded density of the distribution of F(\(\eta)\). As a corollary the rate of convergence of order \(O(n^{-1/2})\) is derived in the functional limit theorem for empirical processes.