CLT implies the sum of n i.i.d random variables,after property normalized converge to a Normal distribution as n goes to infinity. Furthermore, Linderberg's condition points out not necessarily identically distributed ,the one sufficient condition for the sum converges to Normal distribution in the limit case. From a computational perspective, we want to know how numerically the sum converges to a Normal distribution, this is given by Berry-Esseen Theorem. https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem. Essentially, the theorem says the converges rate is determined by the 2,3 moments of those R.V . For convenience,we call this Berry-Esseen metric. B-E metric
With this theoretic results, a natural idea is when summing those R.V ,for some R.V we can well approximate their sums by a normal distribution ,and some 'exotic' r.v for simulation, hence reduce the computation complexity(simulation is time-consuming) .The question is how to build such 'cluster algorithms' Assuming we know 2 and 3rd moments of each R.V? --Noticed that ,unlike tradition Cluster algorithms in Machine learning, where they clustering is based on some 'distance',in this B-E metric, the expression is highly asymmetric for sigma and pho.