On probabilities of large deviations in Banach spaces (Q1124197)

Let \(X,X_ 1,X_ 2,...\in B\) denote a sequence of i.i.d. random variables on a real separable Banach space B, \(Y\in B\) denotes a Gaussian random variable. Suppose that E X\(=E Y=0\) and that the covariances of X and Y coincide. Denote \(S_ n=n^{-1/2}(X_ 1+...+X_ n)\). We prove that under appropriate conditions \[ P(\| S_ n\| >r)=P(\| Y\| >r)(1+o(1))\quad as\quad n\to \infty \] and give estimates of the remainder term. Applications to the \(\omega^ 2\)-test, the Anderson- Darling test and to empirical characteristic functions are given.