being $X, Y$ two continuous processes, $\theta \in R$
I have to show that U and V are independent brownian motions if and only if $X_t$ and $Y_t$ are indenpendent brownian motions.
My approach has been the following...if $X_t$, $Y_t$ are brownian motions then also $U_t$ and $V_t$ are also normal, then I can compute the var/covariance matrix and nullify all the terms but the ones on the main diagonal...is that a good approach? In this case $\theta$ would be $\pi/2$ and X, Y must be independent.
The problem, then, asks me also for a geomtric property of the 2-dim brownian motion that should appear evident from the problem itself...(? can you post a link in which I can go deeper?)