I posted this on Stackexchange already here. Since I got no answer, I wanted to give it a try here (I hope this question is advanced enough).

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}}^2 < \infty$ that is strictly stationary (which means that $W_{n,1},\ldots , W_{n,n}$ is strictly stationary for every $ n$). After Remark 3.3 in "Central Limit and Functional Central Limit Theorems for Hilbert-Valued Dependent Heterogeneous Arrays with Applications" (Chen and White, 1998) the sequence $(\sum_{j=1}^n W_{n,j})_n $ is tight if we have \begin{align*} \lim_{n \rightarrow \infty} \mathbb{E} \bigg\Vert \sum_{j=1}^n W_{n,j} \bigg\Vert_{\mathcal{H}}^2 < \infty. \tag{1} \end{align*} I would like to understand why. It suffices to show (see Lemma 3.2 in the same paper) \begin{align} \lim_{k \rightarrow \infty} \limsup_{n \rightarrow \infty} \mathbb{E} \bigg\Vert \sum_{j=1}^n \sum_{l=k}^{\infty} \langle W_{n,j} , e_l\rangle_{\mathcal{H}} e_l \bigg\Vert_{\mathcal{H}} ^2 =0. \tag{2} \end{align} for some complete orthonormal basis $(e_l)_l$. So far I calculated using straightforward calculations that \begin{align} \mathbb{E} \bigg\Vert \sum_{j=1}^n \sum_{l=k}^{\infty} \langle W_{n,j} , e_l\rangle_{\mathcal{H}} e_l \bigg\Vert_{\mathcal{H}} ^2 = \sum_{i,j=1}^n \mathbb{E} \sum_{l=k}^{\infty} \langle W_{n,j},e_l\rangle_{\mathcal{H}}\langle W_{n,i},e_l\rangle_{\mathcal{H}}. \end{align} Obviously the term within the expectation above is bounded by \begin{align*} \sum_{l=1}^{\infty} \vert \langle W_{n,j},e_l\rangle_{\mathcal{H}}\langle W_{n,i},e_l\rangle_{\mathcal{H}} \vert, \end{align*} which is independent of $k$ and the respective expectations exists by the strict stationarity due to $\mathbb{E} \Vert W_{n,1} \Vert_{\mathcal{H}}^2 < \infty$. Therefore by dominated convergence \begin{align} \lim_{k \rightarrow \infty} \mathbb{E} \bigg\Vert \sum_{j=1}^n \sum_{l=k}^{\infty} \langle W_{n,j} , e_l\rangle_{\mathcal{H}} e_l \bigg\Vert_{\mathcal{H}} ^2 =0. \end{align} Finally the claim would follow if we are able to interchange $\lim_{k \rightarrow \infty}$ and $\limsup_{n \rightarrow \infty}$ in Eq. (2) (probably using cond. (1)), but I do not know how I could verify this. I would really appreciate it if someone could help me with that.