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

Postby vkaul1 » Tue Nov 10, 2009 6:14 pm

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.
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Re: indepence of random variable and sigma algebra

Postby mj » Tue Nov 17, 2009 11:12 pm

This should be true.
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Re: indepence of random variable and sigma algebra

Postby yfrk85 » Fri Dec 02, 2011 8:44 am

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.
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