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](https://projecteuclid.org/euclid.aop/1176995852).) 

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 $\mu.$