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Working Papers (2021), ESSEC Business School

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

KRATZ Marie , PROKOPENKO Evgeny

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

KRATZ, M. and PROKOPENKO, E. (2021). Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence. 2102, ESSEC Business School.

Keywords : #aggregation, #central-limit-theorem, #dependence, #extreme-value-theorem, #geometrical-quantiles, #multivariate-regular-variation, #(multivariate)-Pareto-distribution, #ordered-statistics, #QQ, #plots, #rate-of-convergence, #second-order-regular-variation, #sum-of-random-vectors