For a sum of centered independent random vectors \(X_1, X_2,\ldots\); \(E| X_i|^3 \leq L\), \(E| X_i|^2 \leq 1\) taking values in the space \(\ell_{2k}\) \((k\) is natural) with a common covariance operator and \(Y\) a Gaussian centered random vector in \(\ell_{2k}\) with the same covariance operator the following result is proved: \[ P\{| a+S_n|