Proof that finite-dimensional Wiener process distributions are Gaussian

Question

I have to prove that finite-dimensional Wiener process distributions are Gaussian and calculate them. How should I start? I know the definition and properties of Wiener process.

Answer 1

Hint: For $t_1<t_2<\cdots <t_n$, the vector $$ W=(W_{t_1},W_{t_2}-W_{t_1},W_{t_3}-W_{t_2},\ldots,W_{t_n}-W_{t_{n-1}}) $$ is multivariate Gaussian (why?). Now, use that $$ (W_{t_1},\ldots, W_{t_n})=f(W), $$ where $f(x_1,\ldots,x_n)=(x_1,x_1+x_2,\ldots,x_1+\ldots+x_n)$.