Research output: Contribution to journal › Article › peer-review
Multi-normex distributions for the sum of random vectors. Rates of convergence. / Kratz, Marie; Prokopenko, Evgeny.
In: Extremes, Vol. 26, No. 3, 09.2023, p. 509-544.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 59264541