I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof.
If $X_1, X_2$ and $X_3$ are pairwise independent random variables, then if $Y=X_2+X_3$, is $X_1$ independent to $Y$? (One can think of an example where the $X_i$ s are Bernoulli random variables, then the answer is yes, in the general case I have no idea how to prove it.)
A related problem is:
If $G_1,G_2$ and $G_3$ are pairwise independent sigma algebras, then is $G_1$ independent to the sigma algebra generated by $G_2$ and $G_3$ (which contains all the subsets of both, but has additional sets such as intersection of a set from $G_2$ and a set from $G_3$).
This came about as I tried to solve the following:
Suppose a Brownian motion $\$ is adapted to filtration $\$, if $0<s<t_1<t_2<t_3<\infty$, then show $a_1(W_-W_)+a_2(W_-W_)$ is independent of $F_s$ where $a_1,a_2$ are constants.
By definition individual future increments are independent of $F_s$, for the life of me I don't know how to prove linear combination of future increments are independent of $F_s$, intuitive of course it make sense...
Any help is greatly appreciated.