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: 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.

*This depends on the exact definition of random walk. Surprisingly, I could not find one (at least explicitly) in Hamilton's "Time Series Analysis" (1994) nor in Tsay's "Analysis of Financial Time Series" (3rd edition, 2010) to mention a couple textbooks I checked. I finally found a relevant discussion in Campbell et al. "The Econometrics of Financial Markets" (1997) p. 31-33. They distinguish between three types of random walk:

*

*Type I with i.i.d. increments,

*Type II with independent increments and

*Type III with uncorrelated increments.

Type I is a subset of Type II which in turn is a subset of Type III. The GARCH model above would be an example of the Type III random walk as defined there. I am not sure how common this terminology is in the field of financial econometrics or in other fields. I would not think processes with dependent increments would usually be classified as random walks.
