Previous article Next article On a Local Limit Theorem for Integer Random VectorsN. G. GamkrelidzeN. G. Gamkrelidzehttps://doi.org/10.1137/S0040585X97T987247PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis paper gives an overview of multivariate local limit theorems for integer random vectors and provides the sufficient condition for local limit theorems in ${\bf R}^n$.1. S. N. Bernstein, Some Remarks on Lyapunov Limit Theorem, Vol. 4, Nauka, Moscow, 1964 (in Russian).Google Scholar2. B. V. Gnedenko , On the local limit theorem of the theory of probability , Uspekhi Matem. Nauk , 3 ( 1948 ), pp. 187 -- 190 (in Russian). Google Scholar3. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954.Google Scholar4. P. M. Gruber and K. G. Lekkerkerker, Geometry of Numbers, North--Holland, Amsterdam, 1987.Google Scholar5. J. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin, 1959.Google Scholar6. D. Meizlerbͅ O. Parasyukbͅ and E. Rvacheva , On multidimensional local limit theorem in theory of probability , Ukrainian Math. J. , 1 ( 1949 ), pp. 9 -- 20 (in Russian). Google Scholar7. A. Mitalauskas , On multidimensional limit theorem for lattice distributions , Tr. Lit. SSR Ser. B , 2 ( 1960 ), pp. 3 -- 14 (in Russian). Google Scholar8. V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, New York, Heidelberg, Berlin, 1975.Google Scholar9. Yu. . Prokhorov, On the local limit theorem for lattice distributions , Dokl. Akad. Nauk SSSR , 98 ( 1954 ), pp. 535 -- 538 (in Russian). Google Scholar10. A. Raudelyunas , On the multidimensional local limit theorem , Lit. Matem. Sb. , 4 ( 1964 ), pp. 141 -- 145 (in Russian). Google Scholar11. Yu. . Rozanov, On a local limit theorem for lattice distributions , Theory Probab. Appl. , 2 ( 1957 ), pp. 260 -- 265 . LinkGoogle Scholar12. V. V. Sazonov , On multidimensional central limit theorem , Lit. Matem. Sb. , 1 ( 1963 ), pp. 219 -- 224 (in Russian). Google Scholar13. J. Stoyanov, Counterexamples in Probability, 2nd ed., Wiley, New York, 1997.Google Scholar14. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley, New York, London, Sydney, 1966.Google Scholar15. A. Ya. Khinchin, Mathematical Foundations of Quantum Statistics, Graylock Press, New York, 1960.Google Scholar16. A. Ya. Khinchin, Limit Laws for Sums of Independent Random Variables, ONTI, Moscow, Leningrad, 1938 (in Russian).Google ScholarKeywordsstrong local limit theorem for lattice random variablesasymptotic uniform distribution Previous article Next article FiguresRelatedReferencesCited byDetails Generalization and Refinement of the Integro-Local Stone Theorem for Sums of Random VectorsA. A. Borovkov21 December 2017 | Theory of Probability & Its Applications, Vol. 61, No. 4AbstractPDF (270 KB)A Local Limit Theorem for sums of independent random vectorsElectronic Journal of Probability, Vol. 21, No. none Cross Ref Volume 59, Issue 3| 2015Theory of Probability & Its Applications History Submitted:24 February 2014Published online:01 September 2015 Information© 2015, Society for Industrial and Applied MathematicsKeywordsstrong local limit theorem for lattice random variablesasymptotic uniform distributionPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97T987247Article page range:pp. 494-499ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics