zbMATH — the first resource for mathematics

Upper bounds for minimal distances in the central limit theorem. (English) Zbl 1175.60020
Summary: We obtain upper bounds for minimal metrics in the central limit theorem for sequences of independent real-valued random variables.

60F05 Central limit and other weak theorems
Full Text: DOI EuDML
[1] A. D. Barbour. Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Related Fields 72 (1986) 289-303. · Zbl 0572.60029
[2] P. Bártfai. Über die entfernung der irrfahrtswege. Studia Sci. Math. Hungar. 5 (1970) 41-49. · Zbl 0274.60048
[3] A. Bikelis. Estimates of the remainder term in the central limit theorem. Litovsk. Mat. Sb. 6 (1966) 323-346.
[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. · Zbl 0932.60022
[5] G. Dall’Aglio. Sugli estremi deli momenti delle funzioni di ripartizione doppia. Ann. Sc. Norm. Super Pisa Cl. Sci. 10 (1956) 35-74. · Zbl 0073.14002
[6] C. G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm 121 (1958) 31. · Zbl 0081.35202
[7] M. Fréchet. Recherches théoriques modernes sur le calcul des probabilités, premier livre. Généralités sur les probabilités, éléments aléatoires. Paris, Gauthier-Villars, 1950.
[8] M. Fréchet. Sur la distance de deux lois de probabilité. C. R. Acad. Sci. Paris 244 (1957) 689-692. · Zbl 0077.33007
[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. · Zbl 0161.15207
[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. · Zbl 0308.60029
[11] P. Major. On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8 (1978) 487-517. · Zbl 0408.60028
[12] P. Massart. Strong approximation for multivariate empirical and related processes, via K.M.T. constructions. Ann. Probab. 17 (1989) 266-291. · Zbl 0675.60026
[13] V. V. Petrov. Sums of Independent Random Variables . Berlin, Springer, 1975. · Zbl 0322.60043
[14] E. Rio. Strong approximation for set-indexed partial-sum processes, via KMT constructions I. Ann. Probab. 21 (1993) 759-790. · Zbl 0776.60045
[15] E. Rio. Distances minimales et distances idéales. C. R. Acad. Sci. Paris 326 (1998) 1127-1130. · Zbl 0916.60015
[16] A. I. Sakhanenko. Estimates in the invariance principle. Proc. Inst. Math. Novosibirsk 5 (1985) 27-44. · Zbl 0585.60044
[17] V. M. Zolotarev. On asymptotically best constants in refinements of mean central limit theorems. Theory Probab. Appl. 9 (1964) 268-276. · Zbl 0137.12101
[18] V. M. Zolotarev. Metric distances in spaces of random variables and their distributions. Math. USSR Sbornik 30 (1976) 373-401. · Zbl 0383.60022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.