Let $(M_n)_{n \in \mathbb{N}}$ be a martingale taking values in $\mathbb{R}$, and write
$M_n = A_1 + A_2 + \dots + A_n$
where we do not assume that the $A_i$ are independent. Assume also that
$\mathbb{E}[A_{n+1}^2 \mid \mathcal{F}_n] \leq 1$
for all $n$ almost surely.
Question: Can $\mathbb{P}[M_n \rightarrow -\infty]$ be nonzero?
If we drop the assumption about the bounded variance, the answer is 'yes' - e.g. take
$A_n = \begin{cases} -1 & \text{ with probability } 1-1/n^2 \\ n^2-1 & \text{ with probability } 1/n^2 \end{cases}$
The Chung-Fuchs theorem says that the answer to the question is "no" if $M$ is the sum of iid random variables. But I don't know about the general case.