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Suppose i have $X_1,X_2,...\sim i.i.d. N(0,1)$ $$S_n=X_1+...+X_n$$ filtration $\mathcal{F}_n=\sigma(X_1,...,X_n)$.

$S_n$ is adapted and integrable as finite sum of adapted and integrable random variable. $$E[S_{n+1}|\mathcal{F}_n]=E[S_{n}|\mathcal{F}_n]+E[X_{n+1}|\mathcal{F}_n]=S_n$$ So it's martingale. Can I say that finite sequence $$ \{S_n\}_{i=1}^{n} $$ is martingale or should i have $$ \{S_n\}_{i=1}^{\infty} $$?

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Yes, finite a martingale is a useful thing.

If your particular textbook does not cover the case of a finite martingale $(X_1,X_2,\dots,X_n)$, then extend it by repeating the last value: $(X_1,X_2,\dots,X_n, X_n, X_n, X_n,\dots)$ to get an infinite martingale.

Finite martingales come up, for example, in mathematical finance.

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