I am reading a paper where they prove Donsker's invariance principle for a sequence of dependent RV's. They do the following steps which I can't follow so well. I won't write out the precise definitions since I don't think they are needed, what I don't understand is the logic behind some steps.

Suppose $W_n(t)=n^{-\frac{1}{2}}\sum\limits_{k=1}^{[nt]}X_k$ where $t\in [0,1]$ is the sequence we want to show converges weakly to $W$ where $W$ is our Brownian motion.

This paper looks at a different sequence $M_n$ and shows that $M_n$ converges weakly to $W$. It then says that since for all $T>0$, $\left|\sup\limits_{t\in[0,T]}\left|W_n(t)-M_n(t)\right|\right|_{\mathcal{L}^2}\to 0$, it follows that $W_n\to W$ weakly.

This last step is the one I am struggling to understand. How does the assumption on $M_n$ ensure that $W_n\to W$? It appears to only show the convergence of FDD's (by Slutsky's theorem). Or does this also show tightness somehow? Or am I completely mistaken.

Thanks!

I am reading a paper where they prove Donsker's invariance principle for a sequence of dependent RV's. They do the following steps which I can't follow so well. I won't write out the precise definitions since I don't think they are needed, what I don't understand is the logic behind some steps.

Suppose $W_n(t)=n^{-\frac{1}{2}}\sum\limits_{k=1}^{[nt]}X_k$ where $t\in [0,1]$ is the sequence we want to show converges weakly to $W$ where $W$ is our Brownian motion.

This paper looks at a different sequence $M_n$ and shows that $M_n$ converges weakly to $W$. It then says that since for all $T>0$, $\left|\sup\limits_{t\in[0,T]}\left|W_n(t)-M_n(t)\right|\right|_{\mathcal{L}^2}\to 0$, it follows that $W_n\to W$ weakly.

This last step is the one I am struggling to understand. How does the assumption on $M_n$ ensure that $W_n\to W$? It appears to only show the convergence of FDD's (by Slutsky's theorem). Or does this also show tightness somehow? Or am I completely mistaken.