$2N$ independent random variables. Does the sum of $N$ random variables and the sum of the other $N$ are independent?

Question

$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$

I'm quite sure that it's true, but I don't really know how to prove that.

Answer 1

If $\{X_i:1\leq i \leq n\}\cup \{Y_j\:1\leq j \leq m\}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: \mathbb R^{n} \to \mathbb R$ and $g: \mathbb R^{m} \to \mathbb R$. This follows from the fact that $\sigma (X_1,X_2,...,X_n)$ and $\sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $\sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)\cap X_2^{-1}(A_2)\cap ...\cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $\sigma (Y_1,Y_2,...,Y_m)$].