I got two random processes:
$$y_t=e_t-\frac{1}{3}e_{t-1},\ e_t\sim\mathcal{N}(0,9)\ \text{i.i.d.}$$ $$y_t=e_t-3e_{t-1},\ e_t\sim\mathcal{N}(0,1)\ \text{i.i.d.}$$
I want to show that both have the same family of finite dimensional distributions. But I have no idea how to do that. Can somebody help me?
Since a centered normal random variable with variance $9$ is $3$ times a centered normal random variable with variance $1$, one wants to compare $x_t=3u_t-u_{t-1}$ and $y_t=v_t-3v_{t-1}$ where $(v_t)$ and $(u_t)$ are both i.i.d. standard normal.
Fix some $n\geqslant1$ and some $k$ and consider the random vectors $X=(x_t)_{k+1\leqslant t\leqslant k+n}$ and $Y=(y_t)_{k+1\leqslant t\leqslant k+n}$. Then $X=A(U)$ and $Y=A(S(V))$ where $A$ is (deterministic and) linear, $U=(u_t)_{k\leqslant t\leqslant k+n}$, $V=(v_t)_{k\leqslant t\leqslant k+n}$, and $S$ is the (deterministic) linear involution of $\mathbb R^{n+1}$ defined by $$ S((a_t)_{k\leqslant t\leqslant k+n})=(-a_{n+2k-t})_{k\leqslant t\leqslant k+n}. $$ Since $V$ is i.i.d. with a symmetric distribution, $S(V)$ and $V$ are identically distributed. Since $V$ and $U$ are identically distributed, $X$ and $Y$ are identically distributed. Thus, all the finite marginals of $(x_t)$ and $(y_t)$ coincide, QED.