Error propagation - nonnormal (again) I have a dataset of ~2000 points.  Each of those points has a standard error value associated with it, and it is assumed that the data points and errors are uncorrelated.  Both the dataset and the errors follow a log-normal distribution.   My end-point is to sum all of those data points to get a total value, and get the standard error value for that sum.  Can I use the formula:
$SE_{total} = \sqrt{SE_1^2 + SE_2^2 + \dots + SE_n^2}$ 
even though the distributions are not normal?  I apologize that this is a bit of a repeat of similar questions on this site, but the more I read about this issue the more confused I'm getting!  
 A: The simple answer is YES, you can use the formula and it is exact. If your goal is simply to compute the standard error, then this formula will work for any distributions, as long as your observations are independent.
BONUS.
If you want to go a little further, and try to determine the error distribution, not just the standard error, then it gets a bit complicated.
You should be able to apply the central limit theorem (CLT) in Lyapunov's formulation or Lindberg's. If your errors are truly independent and the variances are finite, then the assumptions of CLT should be applicable. Usually, CLT is assumed to work well for >30 i.i.d. random variables, so 2000 will work exceptionally well. It will be indistinguishable from normal distribution.
One thing to be careful is that Lyapunov (and Lindberg) condition requires that none of the variances are dominating. So, for instance, if one of the $SE_i$ is way-way bigger than the sum of all others, then it will dominate, and your distribution will end up being log-normal. Imagine the error on one of the points is just so huge that all other errors don't matter. It doesn't seem to be the case from your description.
Lyapunov condition is this:$$lim_{n\to \infty}\frac{1}{s_n^3}\sum_iE[|X_i-\mu_i|^3]=0$$
Here's Marlow's paper, which is the main reference on the subject: Bell System Technical Journal, 46: 9. November 1967 pp 2081-2089. A Normal Limit Theorem for Power Sums of Independent Random Variable.
Another paper is Fenton's. He comes up with better approximations for the distribution of the sum of lognormals as a lognormal distribution when $n<<\infty$. 
Again, in the limit, the CLT should be applicable, and normal approximation should work very well, but this is all irrelevant if you simply need to compute standard error only.
