没什么关系啊,
martingale可以是increment independent,也可以不是。举例子的话就是brownian motion is a martingale with increment indenpent,can if we built a process X_t=2*B_t when 0<t<T, and X_t=B_T+B_t when t>=T, this process is gaussian, and you can calculate the correlation and you find that they are not zero.
An independent process is the same, if you take a process like X_t=t, t>0, which is independent increment process, but not a martingale.