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Suppose $X_1, X_2, \dots$ is i.i.d with $P(X_i = 1) = P(X_i = -1) = \dfrac{1}{2}$ and $S_n = X_1+X_+\dots+X_n$. Also, define $N_n = |\{k: S_k = 0,\, 0 \leq k<n\}|$. Then I need to show that $|S_n| - N_n$ is a martingale with respect to the filtration $\mathcal{F}_n = \sigma(S_1,S_2,\dots,S_n)$.

But I am having trouble getting started with showing $E(|S_{n+1}| - N_{n+1}|\mathcal{F}_n) = |S_n| - N_n$. How do I proceed from here?

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Just realized that this question already answers my question perfectly.

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