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Matematicheskie Zametki, 1975, Volume 18, Issue 1, Pages 129–135 (Mi mzm7634)  

This article is cited in 20 scientific papers (total in 21 papers)

An extension of the Lindeberg–Feller theorem

V. I. Rotar'

Central Economics and Mathematics Institute, Academy of Sciences of the USSR
Abstract: A new version is presented of the necessary and sufficient condition of convergence, to a normal law, of sums of independent variables in a nonclassical situation (i.e., absence of limiting negligibility of variables). The obtained condition differs from previously obtained conditions by the fact that it does not use Levy's metric and that is is closer to classical formulations. A similar condition is sufficient for the closeness of two convolutions when the number of components of the convolutions increases without bounds.
Received: 10.11.1974
English version:
Mathematical Notes, 1975, Volume 18, Issue 1, Pages 660–663
DOI: https://doi.org/10.1007/BF01461150
Bibliographic databases:
UDC: 519.21.9
Language: Russian
Citation: V. I. Rotar', “An extension of the Lindeberg–Feller theorem”, Mat. Zametki, 18:1 (1975), 129–135; Math. Notes, 18:1 (1975), 660–663
Citation in format AMSBIB
\Bibitem{Rot75}
\by V.~I.~Rotar'
\paper An extension of the Lindeberg--Feller theorem
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 1
\pages 129--135
\mathnet{http://mi.mathnet.ru/mzm7634}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=400348}
\zmath{https://zbmath.org/?q=an:0348.60025}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 1
\pages 660--663
\crossref{https://doi.org/10.1007/BF01461150}
Linking options:
  • https://www.mathnet.ru/eng/mzm7634
  • https://www.mathnet.ru/eng/mzm/v18/i1/p129
  • This publication is cited in the following 21 articles:
    1. N. G. Gamkrelidze, “Issledovaniya po reshetchatym raspredeleniyam teorii veroyatnostei”, Issledovaniya po reshetchatym raspredeleniyam teorii veroyatnostei, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 218, VINITI RAN, M., 2022, 3–66  mathnet  crossref  mathscinet
    2. “Abstracts of talks given at the 4th International Conference on Stochastic Methods”, Theory Probab. Appl., 65:1 (2020), 121–172  mathnet  crossref  crossref  isi  elib
    3. I. A. Ibragimov, E. L. Presman, Sh. K. Formanov, “On modifications of the Lindeberg and Rotar' conditions in the central limit theorem”, Theory Probab. Appl., 65:4 (2021), 648–651  mathnet  crossref  crossref  isi
    4. E. L. Presman, Sh. K. Formanov, “On Lindeberg–Feller Limit Theorem”, Dokl. Math., 99:2 (2019), 204  crossref
    5. Albert N. Shiryaev, Graduate Texts in Mathematics, 95, Probability-1, 2016, 373  crossref
    6. Daron Acemoglu, Vasco M. Carvalho, Asuman E. Ozdaglar, Alireza Tahbaz-Salehi, “The Network Origins of Aggregate Fluctuations”, SSRN Journal, 2011  crossref
    7. Youri Davydov, Vladimir Rotar, “On a non-classical invariance principle”, Statistics & Probability Letters, 78:14 (2008), 2031  crossref
    8. Sh. K. Formanov, “On the Stein–Tikhomirov method and its applications in nonclassical limit theorems”, Discrete Math. Appl., 17:1 (2007), 23–36  mathnet  crossref  crossref  mathscinet  zmath  elib
    9. Sh. K. Formanov, “The Stein–Tikhomirov Method and a Nonclassical Central Limit Theorem”, Math. Notes, 71:4 (2002), 550–555  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. V. M. Zolotarev, “Natural estimates of convergence rate in the central limit theorem”, J Math Sci, 92:4 (1998), 4112  crossref
    11. Modern Theory of Summation of Random Variables, 1997, 397  crossref
    12. A. N. Shiryaev, Graduate Texts in Mathematics, 95, Probability, 1996, 308  crossref
    13. Hermann Witting, Ulrich Müller-Funk, Mathematische Statistik II, 1995, 1  crossref
    14. A. I. Shtern, B. A. Efimov, S. Yu. Maslov, V. A. Dushskiǐ, P. I. Lizorkin, Yu. A. Bakhturin, I. Kh. Sabitov, A. N. Parshin, A. V. Prokhorov, I. O. Sarmanov, E. D. Solomentsev, V. V. Fedorchuk, V. V. Afanas'ev, E. G. Goluzina, G. V. Kuz'mina, V. V. Sazonov, I. V. Proskuryakov, A. V. Arkhangel'skiǐ, B. V. Khvedelidze, B. I. Golubov, S. A. Telyakovskiǐ, V. A. Chuyanov, V. E. Plisko, P. S. Modenov, A. B. Ivanov, A. S. Fedenko, V. L. Popov, E. M. Chirka, D. P. Zhelobenko, N. N. Vil'yams, A. V. Chernavskiǐ, O. A. Ivanova, G. A. Meshcheryakov, V. I. Pashkovskiǐ, D. D. Sokolov, E. A. Palyutin, M. Sh. Tsalenko, D. V. Anosov, V. A. Skvortsov, V. A. Eleev, L. D. Kudryavtsev, A. M. Nakhushev, V. M. Millionshchikov, A. P. Soldatov, V. V. Pospelov, E. V. Shikin, E. N. Kuz'min, D. B. Anosov, N. K. Nikol'skiǐ, E. G. Sklyarenko, D. O. Baladze, S. N. Malygin, L. A. Skornyakov, Yu. V. Prokhorov, A. L. Onishchik, L. A. Bokut', A. F. A, Encyclopaedia of Mathematics, 1995, 489  crossref
    15. V. I. Rotar', A. G. Sholomitskii, “A Condition for Convergence of Convolutions”, Theory Probab. Appl., 37:2 (1993), 398–401  mathnet  mathnet  crossref
    16. Xing-Hong Xue, “On the principle of conditioning and convergence to mixtures of distributions for sums of dependent random variables”, Stochastic Processes and their Applications, 37:2 (1991), 175  crossref
    17. V. M. Kruglov, “Convergence of distributions of random sums to the normal and the Poisson law”, Theory Probab. Appl., 36:2 (1991), 378–380  mathnet  mathnet  crossref  isi
    18. R. Š. Lipcer, A. N. Širyaev, “On the invariance principle for semimartingales with «nonclassical» assumptions”, Theory Probab. Appl., 28:1 (1984), 1–34  mathnet  mathnet  crossref  isi
    19. V. I. Rotar', “On summation of independent variables in a non-classical situation”, Russian Math. Surveys, 37:6 (1982), 151–175  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    20. V.I Rotar', “Limit theorems for polylinear forms”, Journal of Multivariate Analysis, 9:4 (1979), 511  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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