Showing that a certain stochastic process does not have normal distributed increments 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.
 A: This is the answer he send me regarding these increments.
"As for the increments, $X^{(a)}_e-X^{(a)}_1=
(\sqrt{e}\cos W_{a+1}-\cos W_a)G_1+(\sqrt{e}\sin W_{a+1}-\sin W_a)G_2.$
Since the pair $(W_a, W_{a+1})$ is independent of the pair $(G_1, G_2)$, this
increment has the same distribution as
$[(\sqrt{e}\cos W_{a+1}-\cos W_a)^2+
(\sqrt{e}\sin W_{a+1}-\sin W_a)^{2}]^{1/2} G$,
where $G$ is $\mathcal{N}(0,1)$ independent of the pair $(W_a, W_{a+1})$.
Now, if $Z$ and $G$ are independent, $G$ is $\mathcal{N}(0,1)$-distributed and $ZG$ is
Gaussian then $|Z|$ must be constant a.s. -- indeed, $\mathbb{E}ZG=\mathbb{E}Z \mathbb{E}G=\mathbb{E}Z \cdot 0=0$,
so that $ZG$ is a centered Gaussian random variable; now it suffices to
notice that $\mathbb{E}G^4=(\mathbb{E}G^2)^2, \mathbb{E}(ZG)^4=3(E(ZG)^2)^2$, so that $EZ^4=(EZ^2)^2$ and
hence $Var(Z^2)=0$, i.e., $Z^2$ is constant a.s.
However, the expression standing inside $[...]^{1/2}$ obviously is not
constant a.s., as it is equal to $e+1-2\sqrt{e}\cos(W_{a+1}-W_a)$, and
$W_{a+1}-W_{a}$ is a $\mathcal{N}(0,1)$ Gaussian random variable."
It's a quite impressive and interesting approach.
