Proof of Efron-Stein inequality

Question

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!

Answer 1

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 $\mathbb E[V\mid X_1,\dots,X_j]=0$.