Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, $$E[(X_m-X_l)X_k]=0.$$
2026-04-11 10:46:55.1775904415
Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that
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$X_n$ is an $(\mathcal{F}_n)$-martingale:
$$\mathbb{E}[X_k(X_m-X_l)]=\mathbb{E}[X_k\mathbb{E}[(X_m-X_l)|\mathcal{F}_k]]$$ $$=\mathbb{E}[X_k(X_k-X_k)]=0$$