Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\mathcal{N}(0,\Sigma)$ as $m \to \infty$. As described in Weak convergence for discrete-time processes using characteristic functions, this convergence can be determined via weak convergence of the finite-dimensional marginals.
I'm now interested in the convergence of the Cesaro means of these vectors, i.e. weak convergence of the sequence $\frac{1}{n}\sum_{i=1}^n X_{i,n}$ as $n\to\infty$. Obviously, by the continuous mapping theorem, we know that $\frac{1}{m}\sum_{i=1}^m X_{i,n} \overset{\mathcal{D}}{\to}\mathcal{N}(0,\sigma_m^2)$ with $\sigma^2_m:=\sum_{i,j=1}^m \Sigma_{ij}$ as $n\to\infty$ for any fixed $m$. We also know that $\sigma_m^2 \to \sigma^2 < \infty$.
Question: I would like to show that $\frac{1}{n}\sum_{i=1}^n X_{i,n} \overset{\mathcal{D}}{\to}\mathcal{N}(0,\sigma^2)$. Do I need any additional assumptions to state this convergence or can anybody give me a hint on how to show it?
