TY - JOUR

T1 - Multi-normex distributions for the sum of random vectors. Rates of convergence

AU - Kratz, Marie

AU - Prokopenko, Evgeny

N1 - Evgeny Prokopenko acknowledges the support received from the National Research Agency of the French government through the program “Investment for the future” (ANR-16-IDEX-0008 CY Initiative) during his postdoctoral fellowship at ESSEC CREAR.

PY - 2023/9

Y1 - 2023/9

N2 - 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 multi-normex 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.

AB - 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 multi-normex 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.

KW - (Multivariate) Pareto distribution

KW - 41A25

KW - 60E05

KW - 60F05

KW - 60F15

KW - 60G70

KW - 62G30

KW - Aggregation

KW - Central limit theorem

KW - Dependence

KW - Extreme value theorem

KW - Geometrical quantiles

KW - Multivariate extremes

KW - Multivariate regular variation

KW - Ordered statistics

KW - QQ-plots

KW - Rate of convergence

KW - Second order regular variation

KW - Sum of random vectors

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85146222673&origin=inward&txGid=e09e9fc61ec8a47c06992ad686b37316

UR - https://www.mendeley.com/catalogue/e700a359-d0a3-38ab-bd7a-76f706cd4f06/

U2 - 10.1007/s10687-022-00461-7

DO - 10.1007/s10687-022-00461-7

M3 - Article

VL - 26

SP - 509

EP - 544

JO - Extremes

JF - Extremes

SN - 1386-1999

IS - 3

ER -