Theorem 3.3.2 in KaiLai Chung is as follows:
Let $1 \leq n_1 < n_2 < \dots <n_k = n;$ $f_1$ a Borel measurable function of $n_1$ variables, $f_2$ one of $n_2 - n_1$ variables,..., $f_k$ one of $n_k - n_{k-1}$ variables. If $\{ X_j, 1 \leq j \leq n \}$ are independent r.v.'s then the $k$ r.v.'s $$f_1(X_1,...,X_{n_1}),f_2(X_{n_1+1},...,X_{n_2}),...,f_k(X_{n_k-1},...,X_{n_k})$$
are independent.
My question is: can the above real-valued functions of $f_k$ be extended to general vector-valued real functions? Say $f_k: \mathbb{R}^{n_k} \rightarrow \mathbb{R}^m$, where $m \geq 1$.
here is a proof of a very special case:mutual independence of X, Y, T, Z implies independence of (X+Y, X+T) and Z?
In general, here is the sketch of my proof:
For simplicity, we shall only prove it for $k=2$. Denote $Z_1 = (X_1,...,X_{m})$ and $Z_2=(X_{m+1},...,X_n)$. Then $Y_1 \equiv$ $f_1(Z_1)$ is a $s-dimensional$ random vector and $Y_2 \equiv f_2(Z_2)$ is a $t-dimensional$ random vector. They are independent iff for all $B_1, B_2 \in \mathcal{B}$, we have $$\mathbb{P}(Z_1 \in f_1^{-1}(B_1),Z_2 \in f_2^{-1}(B_2)) = \mathbb{P}(Z_1 \in f_1^{-1}(B_1)\mathbb{P}(Z_2\in f_2^{-1}(B_2))$$ which is implied by the stronger condition $$\mathbb{P}(Z_1 \in A_1, Z_2 \in A_2) = \mathbb{P}(Z_1 \in A_1) \mathbb{P}(Z_2 \in A_2)$$, for all $A_1 \in \mathcal{B}^m$ and $A_2 \in \mathcal{B}^{n-m}.$
To show this, define $\mathcal{B}_1 = \{ A \in \mathcal{B}^m: \mathbb{P}(Z_1 \in A, Z_2 \in B_1 \times ... \times B_{n-m})=\mathbb{P}(Z_1 \in A)\mathbb{P}(Z_2 \in B_1\times...\times B_{n-m}), \text{for any} B_i \in \mathcal{B}, 1 \leq i \leq n-m \}.$
Then it can be shown that:
- The $\pi$-class: $\mathcal{A} = \{A_1\times...\times A_m: A_i \in \mathcal{B}, 1 \leq i \leq m \}$ is contained in $\mathcal{B}_1$.
- $\mathcal{B}_1$ is a $\lambda$-class.
By the $\pi-\lambda$ theorem, we get that $\mathcal{B_1}\supset \sigma(\mathcal{A}) = \mathcal{B}^m.$ In fact, $\mathcal{B}_1 = \mathcal{B}^m$.
Similarly, we can define:
$\mathcal{B}_2 = \{ A \in \mathcal{B}^{n-m}: \mathbb{P}(Z_1 \in B, Z_2 \in A)=\mathbb{P}(Z_1 \in B)\mathbb{P}(Z_2 \in A), \text{for any} B \in \mathcal{B}^m \}.$
In a similar manner, we get $\mathcal{B}_2 = \mathcal{B}^{n-m}.$
That completes the proof.
I think the entire proof builds on the mutual independence of $(X_1,...,X_n)$. It seems to me that whether $f_k$ is real-valued or vector-valued matters litter here, as long as they are Borel measurable.
I am unsure whether I missed some points somewhere in the proof? If this extension is trial, I am curious why textbooks do not adopt this general case?