It is known that a random series 
$$
\sum_{n\geq 1} X_n
$$
whose terms $X_n$ are independent converges  a.s. if and only if it converges in probability.

Is it true that a martingale $(Y_n)$ converges a.s. if and only  if it converges in probability?  If not, are there any counter-examples?   Thanks.