An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

Marcin Dudziński

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2017)

  • Volume: 71, Issue: 1
  • ISSN: 0365-1029

Abstract

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Let: 𝐘 = 𝐘 i , where 𝐘 i = Y i , 1 , . . . , Y i , d , i = 1 , 2 , ⋯ , be a d -dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf F , and F n 𝐱 : = 1 n ∑ i = 1 n 𝕀 Y i , 1 ≤ x 1 , ⋯ , Y i , d ≤ x d denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process B n = n F n - F under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

How to cite

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Marcin Dudziński. "An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 71.1 (2017): null. <http://eudml.org/doc/289767>.

@article{MarcinDudziński2017,
abstract = {Let: $\mathbf \{Y=\}\left( \mathbf \{Y\}_\{i\}\right)$, where $\mathbf \{Y\}_\{i\}=\left( Y_\{i,1\},...,Y_\{i,d\}\right)$, $i=1,2,\dots $, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_\{n\}\left( \mathbf \{x\}\right) :=\frac\{1\}\{n\}\sum _\{i=1\}^\{n\}\mathbb \{I\}\left(Y_\{i,1\}\le x_\{1\},\dots ,Y_\{i,d\}\le x_\{d\}\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_\{n\}=\sqrt\{n\}\left( F_\{n\}-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.},
author = {Marcin Dudziński},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Almost sure central limit theorem; weak dependence; empirical processes; copulas},
language = {eng},
number = {1},
pages = {null},
title = {An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes},
url = {http://eudml.org/doc/289767},
volume = {71},
year = {2017},
}

TY - JOUR
AU - Marcin Dudziński
TI - An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2017
VL - 71
IS - 1
SP - null
AB - Let: $\mathbf {Y=}\left( \mathbf {Y}_{i}\right)$, where $\mathbf {Y}_{i}=\left( Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\dots $, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_{n}\left( \mathbf {x}\right) :=\frac{1}{n}\sum _{i=1}^{n}\mathbb {I}\left(Y_{i,1}\le x_{1},\dots ,Y_{i,d}\le x_{d}\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_{n}=\sqrt{n}\left( F_{n}-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.
LA - eng
KW - Almost sure central limit theorem; weak dependence; empirical processes; copulas
UR - http://eudml.org/doc/289767
ER -

References

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