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

doi:10.1007/978-3-030-76829-4_5

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

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

4 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 language	English (US)
Title of host publication	Operator Theory and Harmonic Analysis, OTHA 2020
Editors	Alexey N. Karapetyants, Igor V. Pavlov, Albert N. Shiryaev
Publisher	Springer
Pages	109-133
Number of pages	25
ISBN (Print)	9783030768287
DOIs	https://doi.org/10.1007/978-3-030-76829-4_5
State	Published - 2021
Event	International Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020 - Rostov-on-Don, Russian Federation Duration: Apr 26 2020 → Apr 30 2020

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	358
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020
Country/Territory	Russian Federation
City	Rostov-on-Don
Period	4/26/20 → 4/30/20

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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10.1007/978-3-030-76829-4_5

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Bobkov, S. G., Danshina, M. A., & Ulyanov, V. V. (2021). Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs. In A. N. Karapetyants, I. V. Pavlov, & A. N. Shiryaev (Eds.), Operator Theory and Harmonic Analysis, OTHA 2020 (pp. 109-133). (Springer Proceedings in Mathematics and Statistics; Vol. 358). Springer. https://doi.org/10.1007/978-3-030-76829-4_5

Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs. / Bobkov, Sergey G.; Danshina, Maria A.; Ulyanov, Vladimir V.
Operator Theory and Harmonic Analysis, OTHA 2020. ed. / Alexey N. Karapetyants; Igor V. Pavlov; Albert N. Shiryaev. Springer, 2021. p. 109-133 (Springer Proceedings in Mathematics and Statistics; Vol. 358).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bobkov, SG, Danshina, MA & Ulyanov, VV 2021, Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs. in AN Karapetyants, IV Pavlov & AN Shiryaev (eds), Operator Theory and Harmonic Analysis, OTHA 2020. Springer Proceedings in Mathematics and Statistics, vol. 358, Springer, pp. 109-133, International Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020, Rostov-on-Don, Russian Federation, 4/26/20. https://doi.org/10.1007/978-3-030-76829-4_5

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AB - 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.

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

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