A PRECISE ESTIMATE OF THE RATE OF CONVERGENCE IN THE CENTRAL-LIMIT-THEOREM IN HILBERT-SPACE

Let [GRAPHICS] be the normed sum of independent identically distributed random variables X(i) with values in a separable Hilbert space H. Denote by V the covariance operator of X, and let Y be an H-valued (0, sigma-2V) Gaussian random variable. The authors prove that there exist an absolute constant c such that for any a is-a-member-of H and r greater-than-or-equal-to O [GRAPHICS] where sigma-1(2) greater-than-or-equal-to sigma-2(2) greater-than-or-equal-to ... are the eigenvalues of V. Up to the value of c, this estimate is unimprovable in general.