# Random walks and martingales

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?

• Hi: I think it would be clearer if you provided the exact definition of random walk type II. Unless of course, the definition is a random walk with innovation terms that are serially uncorrelated but not IID ? – mlofton Jun 18 at 14:46
• exaclty, they are serially uncorrelated but allowed to be dependent. Therefore, a type I is automatically a type II not vice versa. – J3lackkyy Jun 18 at 14:52
• Hi: I'm still slightly confused. Are you referring to the error term, $\epsilon_t$, in $y_t = y_t-1 + \epsilon_t$ ? So, a type II random walk means that the $\epsilon_t$ is not correlated with itself but could be dependent on its previous value in some non-linear way ? If that's correct, then that sounds an ARCH model would fit that criteria. In an ARCH model, the squared error terms are dependent but the error terms themselves are uncorrelated. – mlofton Jun 19 at 15:14
• One other thing: I'm somewhat famiiar with the econometrics literature but I've never heard of type II random walks. You may want to look at "martingale difference sequences" because that could possibly be the same topic but under a different name. Also, I don't think I answered adequately and, since no one else has added anything, you may be better off sending your question to economics.stackexchange.com or quant.stackexchange.com. There may be a terminology issue that's causing confusion. – mlofton Jun 19 at 15:18