Probability theory
On the optimality of McLeish's conditions for the central limit theorem
[Sur l'optimalité des conditions de McLeish pour le théorème limite central]

Présenté par : the Editorial Board

Dedecker, Jérôme1
1 Laboratoire MAP5, CNRS UMR 8145, Université Paris-Descartes, Sorbonne Paris Cité, 45, rue des Saints-Pères, 75270 Paris cedex 06, France
Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 557-561.

Nous construisons une famille de suites strictement stationnaires et ergodiques pour lesquelles le théorème limite central n'a pas lieu. Ces exemples montrent que les conditions de McLeish pour le théorème limite central sont optimales en un sens précis.

We construct a family of stationary ergodic sequences for which the central limit theorem (CLT) does not hold. These examples show that McLeish's conditions for the CLT are sharp in a precise sense.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.03.010
Dedecker, Jérôme 1

1 Laboratoire MAP5, CNRS UMR 8145, Université Paris-Descartes, Sorbonne Paris Cité, 45, rue des Saints-Pères, 75270 Paris cedex 06, France 
@article{CRMATH_2015__353_6_557_0,
     author = {Dedecker, J\'er\^ome},
     title = {On the optimality of {McLeish's} conditions for the central limit theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {557--561},
     publisher = {Elsevier},
     volume = {353},
     number = {6},
     year = {2015},
     doi = {10.1016/j.crma.2015.03.010},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.03.010/}
}
TY  - JOUR
AU  - Dedecker, Jérôme
TI  - On the optimality of McLeish's conditions for the central limit theorem
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 557
EP  - 561
VL  - 353
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2015.03.010/
DO  - 10.1016/j.crma.2015.03.010
LA  - en
ID  - CRMATH_2015__353_6_557_0
ER  - 
%0 Journal Article
%A Dedecker, Jérôme
%T On the optimality of McLeish's conditions for the central limit theorem
%J Comptes Rendus. Mathématique
%D 2015
%P 557-561
%V 353
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2015.03.010/
%R 10.1016/j.crma.2015.03.010
%G en
%F CRMATH_2015__353_6_557_0
Dedecker, Jérôme. On the optimality of McLeish's conditions for the central limit theorem. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 557-561. doi : 10.1016/j.crma.2015.03.010. http://archive.numdam.org/articles/10.1016/j.crma.2015.03.010/

[1] Dedecker, J. Principes d'invariance pour les champs aléatoires stationnaires, Université Paris-Sud, 1998 (Thesis No. 5515)

[2] Dedecker, J.; Merlevède, F. Necessary and sufficient conditions for the conditional central limit theorem, Ann. Probab., Volume 30 (2002), pp. 1044-1081

[3] Dedecker, J.; Merlevède, F.; Volný, D. On the weak invariance principle for non-adapted sequences under projective criteria, J. Theor. Probab., Volume 20 (2007), pp. 971-1004

[4] Gordin, M.I.; Lifšic, B.A. A remark about a Markov process with normal transition operator, Third Vilnius Conference on Probability and Statistics, vol. 1, 1981, pp. 147-148

[5] Hannan, E.J. Central limit theorems for time series regression, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 26 (1973), pp. 157-170

[6] Hannan, E.J. The central limit theorem for time series regression, Stoch. Process. Appl., Volume 9 (1979), pp. 281-289

[7] Heyde, C.C. On the central limit theorem for stationary processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 30 (1974), pp. 315-320

[8] Klicnarová, J.; Volný, D. On the exactness of the Wu–Woodroofe approximation, Stoch. Process. Appl., Volume 119 (2009), pp. 2158-2165

[9] Maxwell, M.; Woodroofe, M. Central limit theorem for additive functionals of Markov chains, Ann. Probab., Volume 28 (2000), pp. 713-724

[10] McLeish, D.L. Invariance principles for dependent variables, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 32 (1975), pp. 165-178

[11] Peligrad, M.; Utev, S. A new maximal inequality and invariance principle for stationary sequences, Ann. Probab., Volume 33 (2005), pp. 798-815

[12] Rényi, A. On stable sequences of events, Sankhya, Ser. A, Volume 25 (1963), pp. 189-206

[13] Volný, D. Martingale approximation and optimality of some conditions for the central limit theorem, J. Theor. Probab., Volume 23 (2010), pp. 888-903

Cité par Sources :