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) |
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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 | |
State | Published - May 20 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 |
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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 |
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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