Brownian motion independent RVs Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that
$W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?

No.
Set $X=W_s(W_t-W_s)$, $Y=W_t(W_l-W_k)$, and $Z=W_k(W_l-W_k)$. Since $Z-Y=(W_k-W_t)(W_l-W_k)$ is independent from $X$, the problem is reduced to prove that $X,Y$ are independent.
We have $\mathbb{E}[X^2]=s(t-s)$, and $\mathbb{E}[Y^2]=t(l-k)$. Then


*

*$\mathbb{E}[X^2Y^2]=3st(t-s)(l-k)$,

*conclude that $X,Y$ are not independent.

