On Lindeberg-Feller Limit Theorem

Cover Page


In the Lindeberg-Feller theorem, the Lindeberg condition is present. The fulfillment of this condition must be checked for any ε > 0. We formulae a new condition in terms of some generalization of moments of order 2 + α, which does not depend on ε, and show that this condition is equivalent to the Lindeberg condition, and if this condition is valid for some α > 0 then it is valid for any α > 0. In the non-classical setting (in the absence of conditions of a uniform infinitely smallness) V.I. Rotar formulated an analogue of the Lindeberg condition in terms of the second pseudo-moments. The paper presents the same modification of Rotar`s condition, which does not depend on ε. In addition, we discuss variants of the simple proofs of theorems of Lindeberg-Feller and Rotar and some related inequalities.

About the authors

E. L. Presman

Central Economics and Mathematics Institute of the Russian Academy of Sciences

Author for correspondence.
Email: presman@cemi.rssi.ru

Russian Federation, 47, Nakhimovsky prospekt, Moscow, 117418

Sh. K. Formanov

Uzbekistan Academy of Sciences V.I.Romanovskiy Institute of Mathematics

Email: shakirformanov@yandex.ru

Uzbekistan, 81, Mirzo Ulugbek street, Tashkent, 100041


  1. Hall P. Rates of Convergence in the Central Limit Theorem. Boston; L.: Pitman Adv. Publ. Progr., 1984. 257 p.
  2. Chen Louis H. Y., Qi-Man Shao. Stein’s Method for Normal Approximation. In: An Introduction to Stein’s Method // Lect. Note. Ser. 4. 2005. P. 1-59.
  3. Ибрагимов И. А., Осипов Л. В. Об оценке остаточного члена в теореме Линдеберга // Теория вероятностей и её применения. 1966. Т. 11. № 1. С. 141-143.
  4. Esseen C. G. On the Remainder Term in the Central Limit Theorem // Arkiv Math. 1968. V. 8. № 1. P. 7-15.
  5. Ротарь В. И. К обобщению теоремы Линдеберга-Феллера // Мат. заметки. 1975. Т. 18. В. 1. С. 129-135.
  6. Ширяев А. Н. Вероятность - 1. М.: МЦНМО, 2004. 520 с.
  7. Formanov Sh.K., Formanova T. A. The Stein-Ti-khomirov Method and Berry-Esseen Inequality for Sampling Sum from a Finite Population of Independent Random Variables. Prokhorov and Contemporary Probability Theory // Springer Proc. Math. and Stat. 2013. V. 33. P. 261-275.



Abstract - 257

PDF (Russian) - 173


Copyright (c) 2019 Russian academy of sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies