In this article, we approach a class of problems in probability theory, namely, the asymptotic expansion of probability. We consider an independent, identically distributed, and normalized stochastic process ()kkX∈ in a separable Hilbert space H, and associate it with the normalized partial sum.

As a result, we built on the ball with a fixed center asymptotic expansion of non-uniform probabilities; our conditions on the moments are minimal, and the dependency of estimates on the covariance operator is expressed with the terms of the eigenvalue series. Likewise, the covariance operators of the random elements do not coincide.

Berry-Esseen, covariance operator, Fourier method, random elements