## indepence of random variable and sigma algebra

This forum is to discuss details in the book "Probability with Martingales" by David Williams.

### indepence of random variable and sigma algebra

I had a question. If random variables X and Y are independent of sigma field \Omega, does that imply that X-Y is also independent of \Omega. More generally is f(X,Y) independent of sigma field \Omega if X and Y are both independent of \Omega?
Intuitively I think it should be the case, but I am unable to prove it.
vkaul1

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Joined: Thu Aug 06, 2009 10:19 pm

### Re: indepence of random variable and sigma algebra

This should be true.
mj
Site Admin

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Joined: Fri Jul 27, 2007 7:21 am

### Re: indepence of random variable and sigma algebra

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.
yfrk85

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Joined: Wed Nov 30, 2011 12:54 pm

### Re: indepence of random variable and sigma algebra

That is sufficient for the proof. I was simply espousing another way to see it in terms of when you distribute a union across the countable intersections.
charlesbabage

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Joined: Sat Apr 18, 2015 6:57 am

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