Geodesic
Exponential inequalities for the tail probabilities of the number of cycles in generalized random graphs
Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 30-43 
A. A. Bystrov; N. V. Volod'ko. Exponential inequalities for the tail probabilities of the number of cycles in generalized random graphs. Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 30-43. http://geodesic.mathdoc.fr/item/MT_2023_26_2_a1/
@article{MT_2023_26_2_a1,
     author = {A. A. Bystrov and N. V. Volod'ko},
     title = {Exponential inequalities for the tail probabilities of the number of cycles in generalized random graphs},
     journal = {Matemati\v{c}eskie trudy},
     pages = {30--43},
     year = {2023},
     volume = {26},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2023_26_2_a1/}
}
TY  - JOUR
AU  - A. A. Bystrov
AU  - N. V. Volod'ko
TI  - Exponential inequalities for the tail probabilities of the number of cycles in generalized random graphs
JO  - Matematičeskie trudy
PY  - 2023
SP  - 30
EP  - 43
VL  - 26
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MT_2023_26_2_a1/
LA  - ru
ID  - MT_2023_26_2_a1
ER  - 
%0 Journal Article
%A A. A. Bystrov
%A N. V. Volod'ko
%T Exponential inequalities for the tail probabilities of the number of cycles in generalized random graphs
%J Matematičeskie trudy
%D 2023
%P 30-43
%V 26
%N 2
%U http://geodesic.mathdoc.fr/item/MT_2023_26_2_a1/
%G ru
%F MT_2023_26_2_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $R_n$ be the centered and normalized number of cycles of fixed length contained in a generalized random graph with $n$ vertices. We obtain a Höffding-type exponential inequality for the tail probability of $R_n$.

[1] I. S. Borisov, A. A. Bystrov, “Eksponentsialnye neravenstva dlya raspredelenii kanonicheskikh protsessov chastichnykh kratnykh summ”, Teoriya veroyatnostei i ee primeneniya, 64:2 (2019), 209–227 | DOI | MR | Zbl

[2] A. V. Logachev, A. A. Mogul'skii, “Eksponentsialnye neravenstva Cheby-sheva dlya sluchainykh grafonov i ikh primenenie”, Sib. matem. zhurn., 61:4 (2020), 880–900 | MR

[3] S. G. Bobkov, M. A. Danshina, V. V. Ulyanov, “Rate of convergence to the Poisson law of the number of cycles in the generalized random graphs. I”, Operator Theory Harmonic Analysis, OTHA 2020, Springer Proc. in Math. and Statist., 358, 2021, 109–132 | MR

[4] A. A. Bystrov, N. V. Volodko, “Exponential Inequalities for the Distribution Tails of the Number of Cycles in the Erdos-Renyi Random Graphs”, Siberian Adv. Math., 32:2 (2022), 87–93 | DOI | MR

[5] A. A. Bystrov, N. V. Volodko, “Exponential inequalities for the number of subgraphs in the Erdos-Renyi random graph”, Statist. Probab. Lett., 195 (2023) | DOI | MR | Zbl

[6] S. Chatterjee, “The missing log in large deviations for triangle counts”, Random Structures Algorithms, 40:4 (2011), 437–451 | DOI | MR

[7] S. Chatterjee, S. R.S. Varadhan, “The large deviation principle for the Erdos-Renyi random graphs”, European J. Combin., 32:7 (2011), 1000–1017 | DOI | MR | Zbl

[8] B. DeMarco, J. Kahn, “Upper tails for triangles”, Publ. Math. Inst. Hungar. Acad. Set, 40:4 (2012), 452–459 | MR | Zbl

[9] P. Erdds, A. Renyi, “On the evolution of random graphs”, Publ. Math. Inst. Hungar. Acad. Sci, 5 (1960), 38–82 | MR

[10] W. Hoffding, “Probability inequalities for sums of bounded random variables”, J. Amer. Statist. Assoc., 58 (1963), 13–30 | DOI | MR | Zbl

[11] S. Janson, K. Oleszkiewicz, A. Rucihski, “Upper tails for subgraph counts in random graphs”, Israel J. Math., 142 (2004), 61–92 | DOI | MR | Zbl

[12] J. H. Kim, V. H. Vu, “Divide and conquer martingales and the number of triangles in a random graph”, Random Structures Algorithms, 24:2 (2004), 166–174 | DOI | MR | Zbl

[13] Q. Liu, Z. Dong, “Limit laws for the number of triangles in the generalized random graphs with random node weights”, Statist. Probab. Lett., 161:1 (2020), 1–6 | DOI | MR

[14] A. Rucihski, When are small subgraphs of a random graph normally distributed?, Probab. Theory Related Fields, 78 (1988), 1–10 | DOI | MR