Expectation conditional on the sum of two random variables

[crguevar]

Hi:

e, z, mu are vectors of size N
I need to show that E(e|z+mu) = E(e|mu) or at least E(e|z+mu) converges in probability to E(e|mu) as N goes to infinity, under the assumption that Z is not correlated with e.

My guess is that to get this result I also need z to be orthogonal to mu, that is z'mu=0

I tryed using the law of iterated expectations... but my bieg problem is that I'm not sure how to handle the condictioning on the sum of z and mu... I would really appreciate any help !

Regards

 

[lippyka]

To show that E(e|z+mu) = E(e|mu), it is enough to assume that Z is not correlated with e. Since Z and mu are both independent of e, we can use the law of iterated expectations to write:E(e|z+mu) = E[E(e|z, mu)|z+mu] = E[E(e|mu)|z+mu] = E(e|mu). This shows that E(e|z+mu) = E(e|mu). To show that E(e|z+mu) converges in probability to E(e|mu) as N goes to infinity, we will need to make additional assumptions about z and mu, such as they being orthogonal or having certain distributions.