In class, our professor explained that the martingale process is the in between case of random walk type I (innovations are i.i.d.) and random walk type II (innovations are serially uncorrelated).
This means that every random walk type I is a martingale but not vice versa, and that every martingale is a random walk type II but not vice versa.
Is it correct that the independence is needed so that the conditional and unconditional expectation is equivalent to satisfy the martingale condition?
Further, what would be an example of a type II random walk that is not a martingale? I am a bit confused how random walk type II processes can be different that one fulfills the martingale condition and one does not. Are there multiple "types" or "cases" of the type II random walk?