On a Central Limit Theorem for Random Elements with Values in Hilbert Space

Abstract

Let $\{ {X_k } \}$ be a sequence of independent random elements with values in a separable Hilbert space H such that ${\bf M}||X_k ||^2 < \infty $, $k = 1,2, \cdots $. For simplicity, assume ${\bf M}X_k = \theta $, $k = 1,2, \cdots $, ($\theta $ is the identity of H) and let $Y_n = \sum\nolimits_{k = 1}^n {X_k } $. Let $\Phi $ be a normal distribution on H with the characteristic functional $\exp \{ { - \tfrac{1}{2}(Sh,h)} \}$, $h \in H$, where S is an S-operator. For a random element X with values in H let $Q_X $ be the distribution on H generated by X and $S_X $ the operator corresponding to the moment matrix of the distribution $Q_X $.A sequence of linear bounded operators $\{ {A_n } \}$ is called a normalizing sequence for the sequence $\{ {X_k } \}$ with respect to the S-operator S if the relation (6) and (7) hold true for it.It is proved that if $\{ {A_n } \}$ is a normalizing sequence for $\{ {X_k } \}$ with respect to S the distributions $Q_{A_n Y_n } $ converge weakly to $\Phi $ and the relation ...