Let $(\Omega, \mathscr{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \dots, X_n, \dots$ be independent real random variables
say if the following proposition are always true or not :
1) $\{ X_1X_2, X_3X_4, \dots, X_nX_{n+1}, \dots\}$ are independent
2) $\{ X_1X_2, X_1X_3, \dots, X_1X_n, \dots\}$ are independent
3) $\sup_{ 1\leq k \leq 15} \sin(X_k^2)$ is independent of $\sum_{k = 18}^{144} e^{X_k + 3X_{k-1}}$
4) $\sum_{k =1}^{10} X_{2k} $ is independent of $\sum_{k =1}^{12} X_{2k+1} $
5) $X_5X_1$ is independent of $X_5$
my work :
1) true, since $\{X_k, X_{k+1}, X_{k+l+1}, X_{k+l+2} \}, \,\,\, k, l \geq1$ are independent then any r.v. which is a function of $X_k, X_{k+1}$ is independent of any r.v. which is a function of $X_{k+l+1}, X_{k+l+2}$
2) true, since :
$$\mathbb{P}(X_1X_k \leq a, X_1X_l \leq b) = \mathbb{P}(A_a, A_b) = \mathbb{P}(A_a \cap A_b) = \mathbb{P}(A_a)\mathbb{P}(A_b) = \mathbb{P}(X_1X_k \leq a)\mathbb{P}( X_1X_l \leq b)$$
where $A_{a (\text{ respectively } b)} = \{ \omega \in \Omega | X_1X_{k (\text{ respectively } l)} \leq a (\text{ respectively } b) \} = \{ \omega \in \Omega | X_{k (\text{ respectively } l)} \in A_{k (\text{ respectively } l)} \in \mathscr{A} \}$
3) true, almost same argument as 1)
4) true, again, same argument as 1)
5) false, take for example $X_1 = 1 \text{ a.s. }$
please tell me if I'm right or if there's anything wrong, thanks.
Well basically if $A, B$ are subsequences of $X_i$ variables, and $f,g$ are (determinstic) functions, then $f(A), g(B)$ are independent if $A, B$ are disjoint. But if $A, B$ are not disjoint, then you cannot conclude $f(A), g(B)$ are independent (they might still be, in specific examples, but you cannot conclude independence in general).
This immediately means independence in cases (1), (3), (4).
For cases (2) and (5), you cannot conclude independence in general, and in fact you can come up with specific examples where the variables involved are dependent.
Your counter-example for (5) is correct. But can you see that almost the same example applies to (2), or at least, the first pair $\{X_1 X_2, X_1 X_3\}$ of (2)?