I´m having some difficulties understanding Donsker´s Invariance Principle following Patrick Billingsley: Convergence of Probability Measures https://notendur.hi.is/ebg6/0471197459.pdf
Here is the context: Let $$X^{n}_{t}(w)= \frac{S_{\lfloor nt\rfloor}(w)}{\sigma \sqrt{n}} + \frac{(nt-\lfloor nt\rfloor)\xi_{\lfloor nt\rfloor+1}}{\sigma \sqrt{n}}$$
where $\xi_1,\xi_2,...$ are independent identically distributed with mean $0$ and variance $0<\sigma^2 < \infty$, $S_{n}= \sum^n_{i=1}\xi_{i}$ and $t\in [0,1]$
Let $0\le s<t\le1$ and consider the bivariate vector $$(X^{n}_{s},X^{n}_{t}-X^{n}_{s})=(\frac{S_{\lfloor ns\rfloor}(w)}{\sigma \sqrt{n}},\frac{S_{\lfloor nt\rfloor}(w)}{\sigma \sqrt{n}}-\frac{S_{\lfloor ns\rfloor}(w)}{\sigma \sqrt{n}})+(\psi_{n,s},\psi_{n,t}-\psi_{n,s})$$
where $$\psi_{n,t} = \frac{(nt-\lfloor nt\rfloor)\xi_{\lfloor nt\rfloor+1}}{\sigma \sqrt{n}}$$ By Central Limit Theorem, Slutsky´s theorem and Chevyshev´s inequality $$X^{n}_{s} \Rightarrow N(0,s)$$ and $$X^{n}_{t}-X^{n}_{s} \Rightarrow N(0,t-s)$$
On page 88 of the book it says that by a similar argument we have that $$(X^{n}_{s},X^{n}_{t}-X^{n}_{s})\Rightarrow (N_1,N_2)$$ where $$N_1,N_2$$ are independent normal with mean $0$ and variances $s$ and $t-s$ respectively. The problem I have is precisely this last statement:
I know that $\frac{S_{\lfloor ns\rfloor}(w)}{\sigma \sqrt{n}}$ is independent of $\frac{S_{\lfloor nt\rfloor}(w)}{\sigma \sqrt{n}}-\frac{S_{\lfloor ns\rfloor}(w)}{\sigma \sqrt{n}}$ for all $n$ and by Central limit theorem they weakly converge to $N_1$ and $N_2$ respectively. But I don´t see why this implies that they jointly converge to $(N_1, N_2)$ and moreover that $N_1$ and $N_2$ are independent
In this link: weak convergence of independent sequence it says that we can have a sequence of random variable such that $X_n \Rightarrow X$ and $Y_n\Rightarrow Y$, $X_n$ independent of $Y_n$ for all $n$ but $X$ and $Y$ may not be independent. So I can´t see why $N_1$ and $N_2$ must be independent
I would really appreciate any hints or suggestions.