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 the triangle inequality, Pythagoras' theorem and monotone convergence 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 \leq \sum_{l=k}^{\infty} \mathbb{E} \langle \sum_{j=1}^n W_{n,j} , e_l\rangle_{\mathcal{H}}^2 \\ \leq \sum_{l=k} n^2 \mathbb{E} \langle W_{n,j},e_l\rangle_{\mathcal{H}}^2, \end{align} where we used the strict stationarity in the last step. Then with applying first monotone convergence and then Parseval's identity we see that this series converges when starting the summation at $l=1$ due to $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}}^2 < \infty$. Therefore \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.