Upper bounds for minimal distances in the central limit theorem
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 802-817.

We obtain upper bounds for minimal metrics in the central limit theorem for sequences of independent real-valued random variables.

Nous obtenons des majorations des distances minimales dans le théorème limite central pour les suites de variables aléatoires réelles indépendantes.

DOI: 10.1214/08-AIHP187
Classification: 60F05
Mots-clés : Fréchet-Dall'Aglio minimal metric, Wasserstein distance, rates of convergence, Esseen's mean central limit theorem, global central limit theorem 
@article{AIHPB_2009__45_3_802_0,
     author = {Rio, Emmanuel},
     title = {Upper bounds for minimal distances in the central limit theorem},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {802--817},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP187},
     mrnumber = {2548505},
     zbl = {1175.60020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/08-AIHP187/}
}
TY  - JOUR
AU  - Rio, Emmanuel
TI  - Upper bounds for minimal distances in the central limit theorem
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 802
EP  - 817
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/08-AIHP187/
DO  - 10.1214/08-AIHP187
LA  - en
ID  - AIHPB_2009__45_3_802_0
ER  - 
%0 Journal Article
%A Rio, Emmanuel
%T Upper bounds for minimal distances in the central limit theorem
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 802-817
%V 45
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/08-AIHP187/
%R 10.1214/08-AIHP187
%G en
%F AIHPB_2009__45_3_802_0
Rio, Emmanuel. Upper bounds for minimal distances in the central limit theorem. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 802-817. doi : 10.1214/08-AIHP187. http://archive.numdam.org/articles/10.1214/08-AIHP187/

A. D. Barbour. Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Related Fields 72 (1986) 289-303. | MR | Zbl

P. Bártfai. Über die entfernung der irrfahrtswege. Studia Sci. Math. Hungar. 5 (1970) 41-49. | MR | Zbl

A. Bikelis. Estimates of the remainder term in the central limit theorem. Litovsk. Mat. Sb. 6 (1966) 323-346. | MR | Zbl

I. S. Borisov, D. A. Panchenko and G. I. Skilyagina. On minimal smoothness conditions for asymptotic expansions of moments in the CLT. Siberian Adv. Math. 8 (1998) 80-95. | MR | Zbl

G. Dall'Aglio. Sugli estremi deli momenti delle funzioni di ripartizione doppia. Ann. Sc. Norm. Super Pisa Cl. Sci. 10 (1956) 35-74. | Numdam | MR | Zbl

C. G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm 121 (1958) 31. | MR | Zbl

M. Fréchet. Recherches théoriques modernes sur le calcul des probabilités, premier livre. Généralités sur les probabilités, éléments aléatoires. Paris, Gauthier-Villars, 1950. | MR | Zbl

M. Fréchet. Sur la distance de deux lois de probabilité. C. R. Acad. Sci. Paris 244 (1957) 689-692. | MR | Zbl

I. Ibragimov. On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables. Theory Probab. Appl. 11 (1966) 559-579. | MR | Zbl

J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent rv's and the sample df. I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131. | MR | Zbl

P. Major. On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8 (1978) 487-517. | MR | Zbl

P. Massart. Strong approximation for multivariate empirical and related processes, via K.M.T. constructions. Ann. Probab. 17 (1989) 266-291. | MR | Zbl

V. V. Petrov. Sums of Independent Random Variables. Berlin, Springer, 1975. | MR | Zbl

E. Rio. Strong approximation for set-indexed partial-sum processes, via KMT constructions I. Ann. Probab. 21 (1993) 759-790. | MR | Zbl

E. Rio. Distances minimales et distances idéales. C. R. Acad. Sci. Paris 326 (1998) 1127-1130. | MR | Zbl

A. I. Sakhanenko. Estimates in the invariance principle. Proc. Inst. Math. Novosibirsk 5 (1985) 27-44. | MR | Zbl

V. M. Zolotarev. On asymptotically best constants in refinements of mean central limit theorems. Theory Probab. Appl. 9 (1964) 268-276. | MR | Zbl

V. M. Zolotarev. Metric distances in spaces of random variables and their distributions. Math. USSR Sbornik 30 (1976) 373-401. | MR | Zbl

Cited by Sources: