Are there non-trivial examples of martingale processes that aren't simple random walks?
I'm trying to better understand the difference between martingales and simple random walks. They look pretty much the same - I've seen people make subtle distinctions between abstract definitions of the two processes, but I'm having some difficulty grasping the difference when presented this way.
While accounts in fair betting/gambling are often cited as examples of martingales, they too appear to be types of simple random walks. Or perhaps someone can help me understand why these examples are definitely not simple random walks. And are there really illuminating examples which might illustrate the difference between these processes? Thanks.