mean of population with repeated samples of varying size - how to apply CLT? I have a population with mean $\mu$ and variance $\sigma^2$.
I draw a sample with $n_i$ number of i.i.d random observations from population. I compute the mean for this sample as $mean_i$. I then repeat this experiment k times drawing samples of different sizes, $n_i$, every time.
Questions:
1) will the $mean_i$ form a normal distribution? (edit: distribution of means across i )
2) What would be the std deviation of such a distribution? It cannot be $\sigma/\sqrt n$ as the n is changing for every sample.
3) If i now take average of all these means, can i construct confidence interval around that measurement for original population mean?
 A: 
Questions: 1) will the mean(i) still form a normal distribution? 

i) The $i^\text{th}$ mean won't be normal (the CLT isn't about samples of size $n_i$), though in very large samples a standardized mean may be close to normally distributed. 
ii) The distribution of means across $i$ won't be normal in general even as sample sizes go to infinity. (I expect this is what you were trying to ask about)

2) What should be the std deviation of such a distribution? 

This is unclear. Are you asking about the distribution of the sample means?
It would be some (scale) mixture distribution, so basic results about the moments - and from that, the variance - of finite mixtures should hold.

3) If i now take average of all these means, can i construct confidence interval around that measurement for original population mean?

Possibly. One example approach: as the number of means goes to infinity, if some conditions hold* you could perhaps make use of an argument relating to the Lyapunov or Lindeberg versions of CLT along with Slutsky's theorem**. There are some other things that might be done.
* see those versions of the CLT; for example, you'll essentially need that none of the variances "dominate" the sum of the variances.
** but again, note that they're actually about what happens in the limit, not at a finite sample size.

Of course, if you know the $n_i$'s and the $\bar{x}_i$'s you can form the unweighted average, and if you have the sample standard deviations, you should also be able to calculate the overall standard deviation as if you had all the data... and you can apply a more "common" approach then.
