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?