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

N. P. Kandelaki   ; V. V. Sazonov

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 1, pp. 43-52

Résumé

Let $\{{X_k}\}$ be a sequence of independent random elements with values in a separable Hilbert space $H$ such that ${\mathbf M}\|X_k\|^2<\infty$, $k=1,2,\dots$ . For simplicity, assume ${\mathbf M}X_k=\theta$, $k=1,2,\dots$, ($\theta$ 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 (8) is true if and only if the condition (9) is satisfied. This result generalizes the Lindeberg-Feller theorem.With a trivial exclusion there always exists a normalizing sequence $\{{A_n}\}$ such that $S_{A_n Y_n}=S^{(n)}$, $S^{(l_n )}$, $l_n\to\infty$, or $S$ (see the definition of $S^{(n)}$ in § 3).