Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

We build a sharp approximation of the whole distribution sum iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends so-called ’normex’ approach from univariate to multivariate framework. propose two possible multinormex distributions, named d-Normex MRV-Normex. Both rely on Gaussian for describing behavior, via CLT, while difference between versions comes using exact or EV theorem maximum. The main theorems provide rate convergence each version multi-normex distributions towards sum, assuming second order regular variation property norm parent vector when considering MRV-normex case. Numerical illustrations comparisons are proposed with various dependence structures vector, QQ-plots based geometrical quantiles.