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Let's consider probability space $(\Omega,\Sigma,P)$ with sequence of random bounded variables $(X_n)$. We assume also that $S_n:=X_1+X_2+...+X_n$. We put filtration $ {\displaystyle {\mathcal {F_n}}}=\Sigma.$

I want to find necessity and sufficiency conditions that S_n is a martingale with respect to filtration $ {\displaystyle {\mathcal {F_n}}}=\Sigma.$

Sufficient Condition

$E(S_{n+1}| {\displaystyle {\mathcal {F_n}}})=E(S_{n}| {\displaystyle {\mathcal {F_n}}})+E(X_{n+1}| {\displaystyle {\mathcal {F_n}}})=S_n+E(X_{n+1}).$

So the Sufficient Condition would be that $S_n$ is $ {\displaystyle {\mathcal {F_n}}}$ measurable and for any $n$ :$E(X_n)=0$.

Is my justification correct ? What can i say about necessity condition ?

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It is not true that $E(X_{n+1}|\mathcal F_n)=EX_{n+1}$. All your random variables are measurable w.r.t. $\mathcal F_n$ so $E(X_{n+1}|\mathcal F_n)=X_{n+1}$. Hence a necessary and sufficient condition for $\{S_n\}$ to be a martinagle is $X_n=0$ almost surely for all $n$.

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