# Are there good examples of martingale processes that are not simple random walks?

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.

A GARCH process with constant conditional mean would be an example of a martingale process that is not a random walk*: \begin{aligned} y_t &= \mu+\varepsilon_t, \\ \varepsilon_t &= \sigma_t z_t, \\ \sigma_t^2 &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\ z_t &\sim i.i.D(0,1) \end{aligned} where $$D$$ is some distribution with zero mean and unit variance. It does not have memory in the conditional mean, so it is a martingale, but it does have memory in the conditional variance, so it is not a random walk.