Checking independent increments in more dimensions Let's say we have a 2-dim. stochastic process $F_{t}:=(F^{1}_{t},F^{2}_{t})$, with components $F^{i}_{t}$. A stochastic process has independent increments, if $\forall n \in \mathbb{N}$ and for every choice of 
$0 \leq t_0 < t_1...< t_n \leq 1$ we have that $F_{t_i}-F_{t_{i-1}} $ where $i=1,2,..n$ are independent.
Question: I want to check if $F_{t}$ has independent increments.
Do we have to check independent increments component-wise i.e. for the first component $F^{1}_{t}$ and for the second component seperately. Or do we have to check also independence between the increments of the first component and the second component; lets say for $t>s>u$ if $F_{t}^{1}-F_{s}^{1}$ is independent of $F_{s}^{2}-F_{u}^{2}$?
In the latter case i would check the covariance matrix $cov[F_{t}-F_{s},F_{s}-F_{u}]$. 

It is not sufficient to consider component wise independence, or even independence between the components. You need exactly the independence that is written in the definition of independent increments: The two-dimensional random variables $F_{t_i} - F_{t_{i - 1}}$ must be independent.
