Consider a collection of equicorrelated Normal random variables: $$ Y \sim N(0, \text{diag}(1-\rho) + \rho\iota_N\iota_N') $$ where $\iota_N$ is a $N$-vector of 1's.
Due to the strong dependence, $\iota_N'Y/\sqrt{N}$ does not converge in distribution. In fact, it is easy to see that $\iota_N'Y/{N} \overset{d}{\to} N(0, \rho).$
Suppose the random variables are equicorrelated but not normal. Can we say anything about the limiting distribution of the mean?
I would also interested in references for limit theorems on "strongly dependent" random variables.