×

zbMATH — the first resource for mathematics

Non-uniform bounds in local limit theorems in case of fractional moments. I. (English) Zbl 1239.60016
Summary: Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order \(s \geq 2\), where \(s\) may be not an integer.

MSC:
60F05 Central limit and other weak theorems
60E10 Characteristic functions; other transforms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions (SIAM edition, 2010) (an updated republication of the book published by Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1986).
[2] B. V. Gnedenko, ”A local limit theorem for densities”, Dokl. Akad. Nauk SSSR (N.S.) 95, 5–7 (1954).
[3] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, translated and annotated by K. L. Chung, with an Appendix by J. L. Doob (Addison-Wesley, Cambridge, Mass., 1954). · Zbl 0056.36001
[4] V. V. Petrov, ”On local Limit Theorems for Sums of Independent Random Variables”, Theory Probab. Appl. 9(2), 312–320 (1964).
[5] V. V. Petrov, Sums of Independent Random Variables (1975). · Zbl 0322.60043
[6] Yu. V. Prohorov, ”A Local Theorem for Densities” Dokl. Akad. Nauk SSSR (N.S.) 83, 797–800 (1952). · Zbl 0046.35301
[7] V. V. Senatov, Central Limit Theorem. Exactness of Approximation and Asymptotic Expansions (TVP Science Publishers, Moscow, 2009) [in Russian].
[8] S. N. Sirazhdinov and M. Mamatov, ”On Mean Convergence for Densities”, Theory Probab. Appl. 7(4), 433–437 (1962). · Zbl 0302.60015
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.