on numerical tests I see that the scaled and centered sum of dependent variates

begin{equation}

{X_1, X_2, …, X_m}hspace{1cm} text{ with } hspace{1cm} X_i sim Bin(n,theta_i), hspace{1 cm }theta_i sim Beta(alpha_i, beta_i)

end{equation}

is normal distributed. For the Ljapunow or Lindeberg version of the CLT it is not a problem that the random variables are not identical distributed. And I guess that the variates meet the requirements. (will check that)

I assume that the dependence can be described as a mixing process. The dependence might vary as it is derived via a minimum spanning tree. I guess the next step would be to find a scheme of random variables and check again if the Ljapunow or the Lindeberg requirement apply.

Could one of you smart guys approve this approach? Or are you experienced with dependencies of the variates with respect to the CLT? Is there a good paper suggestion? Do you have tipps about how create a scheme of random variables? Do you have suggestions which of the both requirements is more easily accessible?

Thanks a lot