On the Rate of Approach of the Distributions of Sums of Independent Random Variables to Accompanying Distributions

Abstract

Previous article Next article On the Rate of Approach of the Distributions of Sums of Independent Random Variables to Accompanying DistributionsI. A. Ibragimov and E. L. PresmanI. A. Ibragimov and E. L. Presmanhttps://doi.org/10.1137/1118092PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. V. Prohorov, On sums of identically distributed random quantities, Dokl. Akad. Nauk SSSR (N.S.), 105 (1955), 645–647, (In Russian.) MR0077008 (17,978h) Google Scholar[2] A. N. Kolmogorov, Two uniform limit theorems for sums of independent random variables, Theory Prob. Applications, 1 (1956), 384–394 10.1137/1101030 LinkGoogle Scholar[3] A. N. Kolmogorov, Approximation of distributions of sums of independent terms by infinitely divisible distributions, Trudy Moskov. Mat. Obšč., 12 (1963), 437–451, (In Russian.) MR0169264 (29:6516) 0134.35401 Google Scholar[4] Lucien Le Cam, J. Neyman and , L. LeCam, On the distribution of sums of independent random variablesProc. Internat. Res. Sem., Statist. Lab., Univ. California, Berkeley, Calif., Springer-Verlag, New York, 1965, 179–202, in Bernoulli, Bayes, Laplace, Berlin-Heidelberg MR0199871 (33:8011) 0139.35203 Google Scholar[5] B. A. Rogozin, An estimate for concentration functions, Theory Prob. Applications, 6 (1961), 94–96 10.1137/1106009 0106.34002 LinkGoogle Scholar[6] C. G. Esseen, On the Kolmogorov-Rogozin inequality for the concentration function, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5 (1966), 210–216 10.1007/BF00533057 MR0205297 (34:5128) 0142.14702 CrossrefGoogle Scholar[7] C. G. Esseen, On the concentration function of a sum of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9 (1968), 290–308 10.1007/BF00531753 MR0231419 (37:6974) 0195.19303 CrossrefGoogle Scholar[8] B. V. Gnedenko and , A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954ix+264, Reading MR0062975 (16,52d) 0056.36001 Google Scholar[9] Yu. P. Studnev, An approximation to the distribution of sums by infinitely divisible laws, Theory Prob. Applications, 5 (1960), 421–425 10.1137/1105044 LinkGoogle Scholar[10] E. L. Presman, On a multi-dimensional version of the Kolmogorov uniform limit theorem, Theory Prob. Applications, 18 (1973), 378–384 10.1137/1118046 0307.60030 LinkGoogle Scholar[11] V. M. Zolotarev, Estimates of the difference between distributions in the Levy metric, Proc. Steklov Inst. Mathem., 112 (1971), 224–231, (In Russian.) 0231.62020 Google Scholar[12] B. V. Gnedenko, Limit theorems for sums of independent random variables, Uspehi Matem. Nauk, 10 (1944), 115–165 MR0012385 (7,19a) Google Scholar[13] Yu. V. Prohorov, Asymptotic behavior of the binomial distribution, Uspehi Matem. Nauk (N.S.), 8 (1953), 135–142 MR0056861 (15,138g) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Infinite Divisibility of InformationIEEE Transactions on Information Theory, Vol. 68, No. 7 Cross Ref On Alternative Approximating Distributions in the Multivariate Version of Kolmogorov's Second Uniform Limit TheoremF. Götze and A. Yu. Zaitsev5 May 2022 | Theory of Probability & Its Applications, Vol. 67, No. 1AbstractPDF (451 KB)On Accompanying Measures and Asymptotic Expansions in the B. V. Gnedenko Limit TheoremV. I. Piterbarg and Yu. A. Scherbakova5 May 2022 | Theory of Probability & Its Applications, Vol. 67, No. 1AbstractPDF (516 KB)О сопровождающих мерах и асимптотических разложениях в предельной теореме Б. В. Гнеденко21 January 2022 | Теория вероятностей и ее применения, Vol. 67, No. 1 Cross Ref Об альтернативных аппроксимирующих распределениях в многомерном варианте второй равномерной предельной теоремы Колмогорова21 January 2022 | Теория вероятностей и ее применения, Vol. 67, No. 1 Cross Ref Infinite Divisibility of Information Cross Ref Infinitely Divisible Approximations for Sums of m-Dependent Random Variables12 July 2017 | Journal of Theoretical Probability, Vol. 31, No. 4 Cross Ref Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables16 June 2018 | Vestnik St. Petersburg University, Mathematics, Vol. 51, No. 2 Cross Ref Definitions and Preliminary Facts17 June 2016 Cross Ref Heinrich’s Method for m-Dependent Variables17 June 2016 Cross Ref Other Methods17 June 2016 Cross Ref The Method of Convolutions17 June 2016 Cross Ref Local Lattice Estimates17 June 2016 Cross Ref Uniform Lattice Estimates17 June 2016 Cross Ref The Esseen Type Estimates17 June 2016 Cross Ref Limit Theorems and Phase Transitions for Two Models of Summation of Independent Identically Distributed Random Variables with a ParameterM. Grabchak and S. A. Molchanov9 June 2015 | Theory of Probability & Its Applications, Vol. 59, No. 2AbstractPDF (247 KB)An expansion in the exponent for compound binomial approximationsLithuanian Mathematical Journal, Vol. 46, No. 1 Cross Ref Infinitely divisible approximations for discrete nonlattice variables1 July 2016 | Advances in Applied Probability, Vol. 35, No. 4 Cross Ref On Compound Poisson Approximations Under Moment RestrictionsV. Cekanavicius25 July 2006 | Theory of Probability & Its Applications, Vol. 44, No. 1AbstractPDF (147 KB)Approximation of Quadratic Forms of Independent Random Vectors by Accompanying LawsV. Bentkus, F. Götze, and A. Yu. Zaitsev17 February 2012 | Theory of Probability & Its Applications, Vol. 42, No. 2AbstractPDF (410 KB)On asymptotic expansions in the first uniform Kolmogorov theoremStatistics & Probability Letters, Vol. 25, No. 2 Cross Ref On smoothing properties of compound poisson distributionsLithuanian Mathematical Journal, Vol. 35, No. 2 Cross Ref On the approximation by convolution of the generalized Poisson measure and the Gaussian distribution. ILithuanian Mathematical Journal, Vol. 32, No. 3 Cross Ref Approximation of sums by compound Poisson distributions with respect to stop-loss distances1 July 2016 | Advances in Applied Probability, Vol. 22, No. 2 Cross Ref On the Uniform Approximation of Distributions of Sums of Independent Random VariablesA. Yu. Zaitsev17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 1AbstractPDF (648 KB)On the Rate of Convergence in Kolmogorov’s Second Uniform Limit TheoremA. Yu. Zaitsev and T. V. Arak17 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 2AbstractPDF (1827 KB)On the Accuracy of Approximation of Distributions of Sums of Independent Random Variables—Which are Nonzero with a Small Probability—By Means of Accompanying LawsA. Yu. Zaitsev28 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 4AbstractPDF (917 KB)On the Convergence Rate in Kolmogorov’s Uniform Limit Theorem. IT. V. Arak17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 2AbstractPDF (1533 KB)Summary Papers Presented at Sessions of the Probability and Mathematical Statistics Seminar at the Leningrad Section of the Mathematical Institute of the USSR Academy of Sciences, 198017 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 3AbstractPDF (957 KB)Two Inequalities for Symmetric Processes and Symmetric DistributionsÉ L. Presman17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 4AbstractPDF (392 KB)On the Approximation of n-Fold Convolutions of Distributions having Non-Negative Characteristic Functions with Accompanying LawsT. V. Arak17 July 2006 | Theory of Probability & Its Applications, Vol. 25, No. 2AbstractPDF (1464 KB)Some Properties of N-Fold Convolutions of DistributionsA. Yu. Zaitsev17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 1AbstractPDF (443 KB) Volume 18, Issue 4| 1974Theory of Probability & Its Applications History Submitted:05 October 1972Published online:28 July 2006 InformationCopyright © 1974 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1118092Article page range:pp. 713-727ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics