Uncorrelated successive differences of martingale

Solution 1:

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

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