I am currently reading these notes and in particular try to understand the proof of theorem 5.12.
Actually, my only questions are about the last paragraph. He claims that $T_n$ is a uniformly decreasing sequence convering to $T$ (so far so good), but then he writes $\mathcal{F}^+(T_n) \subset \mathcal{F}^+(T)$. This does not make sense to me, if we have a decreasing sequence we should have $\mathcal{F}^+(T) \subset \mathcal{F}^+(T_n)$ and I think this is also what he uses then in the independence argument. So we know that $W_{s+T_n} - W_{T_n}$ is independent from $\mathcal{F}^+(T).$
My second problem is, I don't see how all this shows that $Y_t:=W_{t+T}-W_T$ is a Brownian motion. For a Brownian motion, we need to show that continuity of the path holds (okay I see how this could go) and that $Y_t-Y_s \tilde \ \mathscr{N}(0,t-s)$ is also okay to me, but I just don't see how he tries to argue that $Y_t-Y_s$ is independent from $Y_u-Y_v$ where $u<v<s<t$ and I think he somehow mixed that up, as he is talking about the independence of $Y_t$ from $Y_s$ which is not what Brownian motion is about.
Can anybody clarify this point?- If anything is unclear, please let me know.
[Correction: Proof of Theorem 5.13]
It looks like there is a typo in the text, and the claim $\mathcal F^+(T_n)\subset\mathcal F^+(T)$ should read $\mathcal F^+(T)\subset\mathcal F^+(T_n)$. This implies that $\mathcal F^+(T)$ is independent of $\{W_{s+T_n}-W_{T_n}: s\ge 0\}$ for each $n$, hence independent of the limit process $\{W_{s+T}-W_{T}: s\ge 0\}$.
What is $Y_t$? (I don't see it in Leiner's proof of Theorem 5.13.)