Martingale increment proof and conditional expectation

Question

I am reading this lecture note on high dimensional probability and on page 15 it says:

Let $X_1, \ldots, X_n$ be independent random variables and consider a general function $f(x_1, \ldots, x_n)$. Define

$$ \Delta_k = \mathbb{E}[f(X_1, \ldots, X_n)| X_1, \ldots, X_k] - \mathbb{E}[f(X_1, \ldots, X_n)| X_1, \ldots, X_{k-1}].$$

Then $$f(X_1, \ldots, X_n) - \mathbb{E}[f(X_1, \ldots, X_n) = \sum_{k=1}^{n} \Delta_k$$

The note says that $\Delta_1, \ldots, \Delta_k$ are martingale increments by claiming \begin{equation}\label{1} \mathbb{E}[\Delta_k| X_1, \ldots, X_{k-1}] = 0 \qquad (1) \end{equation}

It further claims that, for $l < k$, $$\mathbb{E}[\Delta_k \Delta_k] = \mathbb{E}[\mathbb{E}[\Delta_k|X_1, \ldots, X_{k-1}]\Delta_k] = 0 \qquad (2) $$

While I can informally follow the argument, I am struggling to formally prove (1) and (2) mostly due to confusion with how to handle conditional probability. Could someone help me with how to formally derive (1) and (2)

Answer 1

For (1) this come from the fact that $$\mathbb E\big[\mathbb E[f(X_1,...,X_n)\mid X_1,...,X_k]\mid X_1,...,X_{k-1}\big]=\mathbb E[f(X_1,...,X_n)\mid X_1,...,X_{k-1}]$$ (this is call the tour property of conditional expectation).

For (2) use the fact that $$\mathbb E[\Delta _k\Delta _k]=\mathbb E\big[\mathbb E[\Delta _k\Delta_k\mid X_1,...,X_{k-1}]\big],$$

and remember that $$\mathbb E\big[\mathbb E[f(X_1,...,X_n)\mid X_1,...,X_{k-1}]\mid X_1,...,X_{k-1}\big]=\mathbb E[f(X_1,...,X_n)\mid X_1,...,X_{k-1}].$$

Answer 2

Have a look at the properties of conditional expectation. The one you need to prove (1) is the 'tower property' while for (2) you need 'pulling out known factors' along with the tower property.