Convergence of numerical characteristics of sums of independent random variables with vakues in a Hilbert space

Let $\xi_{n1},\xi_{n2},\dots,\xi_{nm_n}$ be an array of row wise independent random variables with values in a Hilbert space $H$, and let $\varphi$ be a continuous function such that, for any elements $x,y\in H$,
$$ \varphi(x+y)\leq \varphi(x)\varphi(y)\ \text{and}\ \inf_{x\in H} \varphi(x)>0. $$

Assume that $F_n$ (the probability distributions of $\xi_n=\xi_{n1}+\dots+\xi_{nm_n}$) converge weakly to a probability distribution $F$. We prove that
$$ \lim_{n\to\infty}\int_H\varphi(x)F_n(dx)=\int_H\varphi(x)F(dx) $$
if and only if
$$ \lim_{R\to\infty}\sup_n\sum_{j=1}^{m_n}\int_{||x||>R}\varphi(x)F_{nj}^{(s)}(dx)=0, $$
where $F_{nj}$ is the probability distributionof the random variable $\xi_{nj}, F_{nj}^{(s)}=F_{nj}*\overline{F}_{nj}$, $\overline{F}_{nj}(A)=F_{nj}(-A)$.
Some results are derived from this theorem.