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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?

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.

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.

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?