Edit: Question Resolved. See below.
As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process.
Let $G_1, G_2, W_t$ be independent. Here we have $G_1$, $G_2$ $\sim \mathcal{N}(0,1)$ and $W_t$ Brownian motion. For $a\geq 0$ and $t\geq e^{-a}$ define the filtration $\mathcal{F}_t^{(a)}=\sigma(G_1,G_2,(W_s)_{0\leq s\leq a+\ln t}),$ and the process $$X_t^{(a)} = \sqrt{t}\left(G_1\cos{W_{a+\ln t}}+ G_2\sin{W_{a+\ln t}}\right).$$
Now Oleskiewicz claims,
(1) $X_t^{(a)}$ is a continuous martingale,
(2) $X_t^{(a)}\sim \mathcal{N}(0,t)$,
(3) $X_e^{(a)}-X_1^{(a)}$ is not gaussian,
without actually proofing any of these statements.
While I managed to proof the first two statements, the proofs are rather long and involved, so I wanted to ask if I am missing something obvious. Furthermore, I am absolutely clueless regarding the third statement.
I hope you can help me.