I am trying to understand the diffrence between random walk and martingale. According to my understanding, a random walk without drift is
$$ y_{t} = y_{t-1} + u_{t} $$
where $u_{t}$ is $i.i.d.(0, \sigma_{t}^2)$ where $\text{Cov}(y_{t},y_{t-s})=0$ for $t \neq s$.
However, a martingale has just one restiction:
$$ \mathbb{E}[y_{t}|y_{t-1},y_{t-2},\dotsc] = y_{t-1} $$
and there are no restrictions on higher order moments. The distribution of $u_{t}$ is just required to satisfy $\mathbb{E}(u_{t})=0$. Heterogeneity of $u_{i}$ and/or correlation between sequences of $u_{i}$ are allowed.
Is my understanding right?