I wonder whether Theorem 2 from the paper J. Zinn, Annals of Probability, 1977, vol. 5, 283-286 can be extended to the CLT for a scheme of series. (The paper is available in the web.)

Let $G$ be a Banach space. Suppose that we have a triangular array $X_{ik}, 1\le i \le k$ of independent in each fixed row $G$-valued random elements and want to have the CLT for such array. Let a Radon probability $\mu^i$ be the distribution of entries is $i$-th row. Let for each $i, \mu^i$ satisfy conditions (b) and (c) of Theorem 2, and the second moments of $\mu^i$ from condition (a) be uniformly bounded. Moreover suppose that $\mu^i$ converges weakly to $\mu$. Does it imply (perhaps under additional assumptions) that the CLT for the array holds true? That is $\Sigma_{i=1}^k X_{ik} / \sqrt{k}$ converges in distribution to $\mu.$