Expansion Of Sum Research Articles

The Central Limit Theorem for Sums of Functions of Independent Variables and Sums of the Form $\sum {f(2^k t)} $

Previous article Next article The Central Limit Theorem for Sums of Functions of Independent Variables and Sums of the Form $\sum {f(2^k t)} $I. A. IbragimovI. A. Ibragimovhttps://doi.org/10.1137/1112075PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. G. Postnikov, Ergodic aspects of the theory of congruences and of the theory of Diophantine approximations, Trudy Mat. Inst. Steklov, 82 (1966), 3–112, (In Russian.) MR0214561 0178.04702 Google Scholar[2] I. A. Ibragimov and , Yu. V. Linnik, Independent and Stationarily Related Variables, Izd-vo “Nauka”, Moscow, 1965, (In Russian.) Google Scholar[3] S. N. Bernstein, Collected Works, Vol. 4, Izd-vo “Nauka”, Moscow, 1965, (In Russian.) Google Scholar[4] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963xvi+685 MR0203748 0108.14202 Google Scholar[5] G. H. Hardy, , J. E. Littlewood and , G. Pólya, Inequalities, Cambridge, at the University Press, 1952xii+324, 2nd ed., London MR0046395 0047.05302 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent VariablesVestnik St. Petersburg University, Mathematics, Vol. 51, No. 3 | 4 September 2018 Cross Ref Berry–Esseen theorems under weak dependenceThe Annals of Probability, Vol. 44, No. 3 | 1 May 2016 Cross Ref Split invariance principles for stationary processesThe Annals of Probability, Vol. 39, No. 6 | 1 Nov 2011 Cross Ref On mean central limit theorems for stationary sequencesAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 44, No. 4 | 1 Aug 2008 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Le théorème limite central pour des sommes de Riesz-RaikovProbability Theory and Related Fields, Vol. 93, No. 4 | 1 Dec 1992 Cross Ref From dynamical systems to the Langevin equationPhysica A: Statistical Mechanics and its Applications, Vol. 145, No. 1-2 | 1 Sep 1987 Cross Ref A local limit theorem for a sequence of interval transformationsErgodic Theory and Dynamical Systems, Vol. 5, No. 2 | 19 September 2008 Cross Ref Asymptotic expansions for sums of weakly dependent random vectorsZeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 64, No. 2 | 1 Jun 1983 Cross Ref Summary of Reports Presented at Sessions of the Probability and Statistics Seminar in the Leningrad Section of the Mathematical Institute of the USSR Academy of Sciences, 1979Theory of Probability & Its Applications, Vol. 26, No. 2 | 17 July 2006AbstractPDF (768 KB)A Non-Uniform Estimate of the Rate of Convergence in the Central Limit Theorem for the Sums of Vector-Valued Functions of Independent VariablesR. M. Gil’fanovTheory of Probability & Its Applications, Vol. 26, No. 4 | 17 July 2006AbstractPDF (830 KB)The Central Limit Theorem for Sums of Functions of Mixing SequencesV. T. Dubrovin and D. A. MoskvinTheory of Probability & Its Applications, Vol. 24, No. 3 | 17 July 2006AbstractPDF (701 KB)A Local Limit Theorem for the Distribution of Fractional Parts of an Exponential FunctionD. A. Moskvin and A. G. PostnikovTheory of Probability & Its Applications, Vol. 23, No. 3 | 17 July 2006AbstractPDF (548 KB)An Estimate for the Remainder Term in the Central Limit Theorem for Sums of Functions of Independent Variables and Sums of the Form $\sum {f(t2^k )} $V. I. Ladokhin and D. A. MoskvinTheory of Probability & Its Applications, Vol. 16, No. 1 | 17 July 2006AbstractPDF (542 KB)On the Asymptotic Behavior of Large Deviation Probabilities for the Sums $\Sigma f(x 2^n )$D. A. MoskvinTheory of Probability & Its Applications, Vol. 15, No. 2 | 17 July 2006AbstractPDF (636 KB)The Central Limit Theorem for Some Weakly Dependent SequencesV. F. GaposhkinTheory of Probability & Its Applications, Vol. 15, No. 4 | 17 July 2006AbstractPDF (1143 KB) Volume 12, Issue 4| 1967Theory of Probability & Its Applications535-705 History Submitted:15 December 1966Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1112075Article page range:pp. 596-607ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Infrared contributions to intermediate state sums in quantum electrodynamics

A method has been developed for incorporating the Bloch-Norsdsieck approximation into intermediate state sum expansions in quantum electrodynamics, so as to explicitly include the contributions of soft photons to these sums. This accomplishes the removal of infrared divergences usually incurred in approximations involving the cutting off of such intermediate state sums, and in addition, within the framework of such approximations, gives strong high energy convergence properties to the theory. The formalism developed is applied to modify Fainberg's work on quantum electrodynamic dispersion relations (1). An asymptotic form for the electron's charge form factor is derived, and is shown to be consistent with second order perturbation theory, and a series expansion for the charge renormalization constant Z 3 is obtained, all of whose terms are finite, except for those which arise from states containing only hard photons and no pairs. In a separate development, a comparison is made of the various methods which have been developed for treating the radiative corrections to scattering cross sections due to soft photons.

Some Limit Theorems for Large Deviations

Some Limit Theorems for Large Deviations