Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs

Sergey G. Bobkov, Maria A. Danshina, Vladimir V. Ulyanov

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

Convergence of orderO(1∕n) is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is based on the Chen–Stein approach and on the derived properties of the ratio of the sum of squares of random variables and the sum of these variables. These properties can be applied to other asymptotic problems related to generalized random graphs.

Original languageEnglish (US)
Title of host publicationOperator Theory and Harmonic Analysis, OTHA 2020
EditorsAlexey N. Karapetyants, Igor V. Pavlov, Albert N. Shiryaev
PublisherSpringer
Pages109-133
Number of pages25
ISBN (Print)9783030768287
DOIs
StatePublished - May 20 2021
EventInternational Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020 - Rostov-on-Don, Russian Federation
Duration: Apr 26 2020Apr 30 2020

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume358
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceInternational Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020
Country/TerritoryRussian Federation
CityRostov-on-Don
Period4/26/204/30/20

Bibliographical note

Funding Information:
Acknowledgments Theorem 1 has been obtained under support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2019-1621. Theorems 2 and 3 were proved within the framework of the HSE University Basic Research Program. Research of S.Bobkov was supported by the NSF grant DMS-1855575.

Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Keywords

  • Cycles
  • Generalized random graphs
  • Poisson law
  • Rate of convergence

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