I read somewhere that given a martingale ${X_n}$, the successive differences of the martingale series are uncorrelated, namely $X_i −X_{i−1}$ is uncorrelated with $X_j −X_{j−1}$ for $i \neq j$. I tried showing this with the law of total covariance but didn't seem to get anywhere. Any thoughts on how to proceed with this problem?


Denote by $\mathcal{F}_n := \sigma(X_1,\ldots,X_n)$ the canonical $\sigma$-algebra and fix $i<j$. Then,

$$\mathbb{E}(X_j - X_{j-1} \mid \mathcal{F}_i) = \mathbb{E}(X_j \mid \mathcal{F}_i)- \mathbb{E}(X_{j-1} \mid \mathcal{F}_i) = X_i-X_i = 0. \tag{1}$$

Using the tower property and pull out, we get

$$\begin{align*} \mathbb{E}[(X_i-X_{i-1}) \cdot (X_j-X_{j-1})] &= \mathbb{E} \bigg[ \mathbb{E}((X_i-X_{i-1}) \cdot (X_j-X_{j-1}) \mid \mathcal{F}_i) \bigg] \\ &= \mathbb{E} \bigg[ (X_i-X_{i-1}) \underbrace{\mathbb{E}( X_j-X_{j-1} \mid \mathcal{F}_i)}_{\stackrel{(1)}{=}0} \bigg] =0 \end{align*}$$


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