Proof of Efron-Stein inequality I was going through the proof of Efron-Stein inequality in this paper (page 219). I was unable to understand the following step $$\mathbb E\left[V_j  \mathbb E\left[V_i|X_1,...,X_j\right]\right] = 0 \text{ for } i>j.$$
It is not clear to me how $\mathbb E\left[V_i|X_1,...,X_j\right] = 0$ for $i>j$. 
Thanks in advance!

Notice that $j\leqslant i-1$, hence by the towering property, 
$$\mathbb E\left[\mathbb E\left[Z\mid X_1,\dots,X_{i-1}  \right]\mid X_1,\dots,X_j\right]=\mathbb E\left[Z\mid X_1,\dots,X_j \right].$$
By the same argument, 
$$\mathbb E\left[\mathbb E\left[Z\mid X_1,\dots,X_{i}  \right]\mid X_1,\dots,X_j\right]=\mathbb E\left[Z\mid X_1,\dots,X_j \right],$$
hence substracting the two equations, we get $\mathbb E[V\mid X_1,\dots,X_j]=0$. 
