A new convergence problem for the conditional expectation You have risks $X_1$, $X_2$, ...
(they are assumed to be independent, but not necessarily identically distributed)
and
$S_n= X_1 + X_2 + \cdots +X_n$
QUESTION: under what reasonable conditions do we have that
$\mathbb{E}[Xi | S_n]$ converges to $\mathbb{E}[Xi]$?
 A: Let's assume that all the $X_i$ have finite mean and variance (not necessarily uniformly bounded at this point).
I claim it is necessary that $\mathrm{var}[S_n]\to\infty$, since otherwise the correlation between $X_i$ and $S_n$ doesn't go to zero.
This is not sufficient. Suppose all the $X_i$ take on only even values, except that $X_1$ is 0 or 1 with equal probability.  If we see $S_n$ is odd, we know $X_1$ is odd and therefore $X_1=1$. If we see $S_n$ is even we know $X_1$ is even and therefore 0. So it is not the case that $E[X_1|S_n]\to E[X_1]$. You could probably do other clever things taking $X_i$s to be 0 or a logarithm of a prime to get it to fail for more different $X_i$ at once.
I think the variance going to infinity is sufficient except when you're cheating like I was with the odd/even numbers.  That is, I think something like the following is true: the result holds if  you're allowed to add noise to stop any trickery.
If $\mathrm{var}[S_n]\to\infty$ then $E[\tilde X_i|S_n]\to E[\tilde X_i]$ for $\tilde X_i=X_i+\epsilon_i$ for any iid finite-variance absolutely continuous $\epsilon_i$
