Difference between stationarity and independence properties for Brownian motion What is the difference between the stationarity and independence properties of proving that a stochastic process $W(t)$ is Brownian motion?
I only understand that for stationarity, we're trying to show that $W(t+\Delta t) - W(t)$ is independent of $t$ and at best only dependent on $\Delta t$. How do I show independence then or how do I intuitively understand independence? 
 A: I assume that you already know both definitions, so you actually know that both definitions aren't the same and furthermore none is implying the other, which means:


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*stationary increments $\not \Rightarrow$ independent increments


For example check this answer, where @Did is constructing an example to match this case.


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*independent increments $\not \Rightarrow$ stationary increments


For this to see, just have a look at the non-homogeneous Poisson process, which indeed has independent increments but is not necessarily stationary.
Stationarity means basically, that the distributional properties of $W_t-W_s$ only depend on $(t-s)$, so that  $W_t-W_s \sim W_{t-s}$.
On the other hand independent increments means, that knowing the increment of $W_{t_1}-W_{t_0}$ gives you no distributional information of the increment of any other non-overlapping increment like $W_{t_2}-W_{t_1}$ given $t_0<t_1<\ldots <t_n$.
Form some more intuition on this I recommend to also check some of these comments.
