Upper bounds for minimal distances in the central limit theorem

Emmanuel Rio

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 802-817
  • ISSN: 0246-0203

Abstract

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

How to cite

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Rio, Emmanuel. "Upper bounds for minimal distances in the central limit theorem." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 802-817. <http://eudml.org/doc/78045>.

@article{Rio2009,
abstract = {We obtain upper bounds for minimal metrics in the central limit theorem for sequences of independent real-valued random variables.},
author = {Rio, Emmanuel},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Fréchet–Dall’Aglio minimal metric; Wasserstein distance; rates of convergence; Esseen’s mean central limit theorem; global central limit theorem; Fréchet-Dall’Aglio minimal metric; Esseen's mean central limit theorem},
language = {eng},
number = {3},
pages = {802-817},
publisher = {Gauthier-Villars},
title = {Upper bounds for minimal distances in the central limit theorem},
url = {http://eudml.org/doc/78045},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 802
EP - 817
AB - We obtain upper bounds for minimal metrics in the central limit theorem for sequences of independent real-valued random variables.
LA - eng
KW - Fréchet–Dall’Aglio minimal metric; Wasserstein distance; rates of convergence; Esseen’s mean central limit theorem; global central limit theorem; Fréchet-Dall’Aglio minimal metric; Esseen's mean central limit theorem
UR - http://eudml.org/doc/78045
ER -

References

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  1. A. D. Barbour. Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Related Fields 72 (1986) 289–303. Zbl0572.60029MR836279
  2. P. Bártfai. Über die entfernung der irrfahrtswege. Studia Sci. Math. Hungar. 5 (1970) 41–49. Zbl0274.60048MR275499
  3. A. Bikelis. Estimates of the remainder term in the central limit theorem. Litovsk. Mat. Sb. 6 (1966) 323–346. Zbl0149.14002MR210173
  4. 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. Zbl0932.60022MR1651906
  5. G. Dall’Aglio. Sugli estremi deli momenti delle funzioni di ripartizione doppia. Ann. Sc. Norm. Super Pisa Cl. Sci. 10 (1956) 35–74. Zbl0073.14002MR81577
  6. C. G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm 121 (1958) 31. Zbl0081.35202MR97111
  7. M. Fréchet. Recherches théoriques modernes sur le calcul des probabilités, premier livre. Généralités sur les probabilités, éléments aléatoires. Paris, Gauthier-Villars, 1950. Zbl0038.08304MR38581
  8. M. Fréchet. Sur la distance de deux lois de probabilité. C. R. Acad. Sci. Paris 244 (1957) 689–692. Zbl0077.33007MR83210
  9. 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. Zbl0161.15207MR212853
  10. 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. Zbl0308.60029MR375412
  11. P. Major. On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8 (1978) 487–517. Zbl0408.60028MR520959
  12. P. Massart. Strong approximation for multivariate empirical and related processes, via K.M.T. constructions. Ann. Probab. 17 (1989) 266–291. Zbl0675.60026MR972785
  13. V. V. Petrov. Sums of Independent Random Variables. Berlin, Springer, 1975. Zbl0322.60042MR388499
  14. E. Rio. Strong approximation for set-indexed partial-sum processes, via KMT constructions I. Ann. Probab. 21 (1993) 759–790. Zbl0776.60045MR1217564
  15. E. Rio. Distances minimales et distances idéales. C. R. Acad. Sci. Paris 326 (1998) 1127–1130. Zbl0916.60015MR1647215
  16. A. I. Sakhanenko. Estimates in the invariance principle. Proc. Inst. Math. Novosibirsk 5 (1985) 27–44. Zbl0585.60044MR821751
  17. V. M. Zolotarev. On asymptotically best constants in refinements of mean central limit theorems. Theory Probab. Appl. 9 (1964) 268–276. Zbl0137.12101MR163338
  18. V. M. Zolotarev. Metric distances in spaces of random variables and their distributions. Math. USSR Sbornik 30 (1976) 373–401. Zbl0383.60022MR467869

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