Law of total expectation in the Efron-Stein inequality

Question

I was reading the Efron-Stein inequality (see for example). To get the context, define $X_1,...,X_n$, a set of iid random variables and $Z=f(X_1,...,X_n)$.

In one step of the proof one uses that for $j>i$, $$ E[Z|X_1,...,X_i] = E[\,E[Z|X_1,...,X_j]\,|\,X_1,...,X_i]. $$ I understand that the equality is true by the Law of Total Expectation. But, why is it relevant that $j>i$? Why that equality doesn't work for $j\leq i$?

Answer 1

Hint:

Tower property

If $\mathcal F_1 \subset \mathcal F_2$ so $\mathbb E(\mathbb E(Z\mid \mathcal F_2)\mid \mathcal F_1)=\mathbb E(\mathbb E(Z\mid \mathcal F_1)\mid \mathcal F_2)=\mathbb E(Z|\mathcal F_1).$

Since $j>i$ so $\mathcal F_1=\sigma(X_1,\cdots , X_i)\subset \mathcal F_2=\sigma(X_1,\cdots , X_j).$