Central limit theorem for sum from varied distributions The central limit theorem as I am familiar with it applies to the limiting (rescaled) distribution of $n$ convolutions of a single probability distribution as $n$ goes to infinity, or equivalently, to distribution one gets from taking a sum of $n$ random variables each with a single fixed distribution. That is, it is a theorem about the (limiting as $n\to \infty$) probability distribution of
$A_1 + A_2 + ... + A_n$ where each term has a fixed distribution $P$.
I am asking about a theorem about the limiting probability distribution of
$A_1 + A_2 + ... + A_n$
where $A_1$ has probability distribution $P_1$, $A_2$ has probability distribution $P_2$, $A_3$ has probability distribution $P_3$, etc.
Also, is there a theorem for the case where each distribution isn't fixed, but is selected at random with probability determined by a measure $\mu$?
Is there such a general theorem, where the limit isn't necessarily gaussian, the limit can be reconstructed from $\mu$, and the convergence is pretty strong?
 A: The theorem 3.1 in this book answers your first question. The key restriction in central limit theorem is not different distributions but the independence. The result is a very nice one, since it says that for interesting sums of  independent random variables the limiting distribution has to have certain property, namely infinite divisibility. The classical central limit theorem (with iid variables with finite variances) is then only a very special case of this theorem. 
Note that this is a very general answer to very general question. Given the nature of your distributions more precise answer can be given. For example if the distributions  satisfy Lindeberg's condition then the limiting distribution is necessary normal (if we exclude let us say non-interesting cases). 
A: mpiktas gave a very good technical answer.
There's a nice simulation of this here:
http://onlinestatbook.com/stat_sim/sampling_dist/index.html 
You can manipulate the distribution at the top of this demo to show a combination of different distributions (such as bimodal), and the distribution of sample means will still be normal.
