Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space

We consider short asymptotic expansions for the probability of a sum of i.i.d. random elements to hit a ball in a Hilbert space H. The error bound for the expansion is of order O(n^-1). It depends on the first 12 eigenvalues of the covariance operator only. Moreover, the bound is non-uniform, i.e. the accuracy of the approximation becomes better as the distance between a boundary of the ball and the origin in H grows.