Weak convergence in Skorohod space

Question

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.

Answer 1

Please use following results in Billingsley's book: Convergence of Probability Measures, 2ed. p.124 Th3.1 p.27.
Theorem Suppose that $(X_n,Y_n)$ are random elements of $S\times S$. If $X_n\Rightarrow X$ and $\rho(X_n,Y_n)\Rightarrow 0$, then $Y_n\Rightarrow Y$.
Write $$ W_n(t)=M_n(t)+[W_n(t)-M_n(t)], $$ Then i) $M_n\stackrel{w}{\to}W$ in Skorokhod topology;

ii) $\mathsf{P}(\rho(W_n,M_n)>\varepsilon)\to0$, $\forall \varepsilon>0$, since $\mathsf{E}[\sup_t|W_n(t)-M_n(t)|^2]\to 0$ and $\mathsf{P}(\sup_t|W_n(t)-M_n(t)|>\varepsilon)\to0$, $\forall \varepsilon>0$.
Now by using Th.3.1 and from this two facts it is ready to get $W_n\stackrel{w}{\to}W$ in Skorokhod topology.