Geodesic
On the sizes of trees in a Galton–Watson forest with infinite variance in the critical case
Diskretnaya Matematika, Tome 36 (2024) no. 2, pp. 33-49 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Galton-Watson forests formed by a critical branching process starting with $N$ particles are considered. The total number of descendants of the initial particles is equal to $n$ for all the time of evolution. Assume that the number of offspring of each particle has the distribution $$p_k=\frac{h(k+1)}{(k+1)^\tau}, \quad k=0,1,2, \ldots, \quad \tau\in (2,3),$$ where $h(x)$ is a slowly varying at infinity function. The generating function of this distribution has the form $$U(z)=z+(1-z)^{\tau-1}L(1-z),$$ where $L(x)$ is a slowly varying function as $x\rightarrow 0.$ The limit distributions of the number of trees of a given size are found as $N,n\rightarrow \infty$ such that $n/N^{\tau-1} \rightarrow \infty.$
Keywords: Galton–Watson forest, tree size, limit theorems. 
@article{DM_2024_36_2_a3,
     author = {Yu. L. Pavlov},
     title = {On the sizes of trees in a {Galton{\textendash}Watson} forest with infinite variance in the critical case},
     journal = {Diskretnaya Matematika},
     pages = {33--49},
     year = {2024},
     volume = {36},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2024_36_2_a3/}
}
TY  - JOUR
AU  - Yu. L. Pavlov
TI  - On the sizes of trees in a Galton–Watson forest with infinite variance in the critical case
JO  - Diskretnaya Matematika
PY  - 2024
SP  - 33
EP  - 49
VL  - 36
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DM_2024_36_2_a3/
LA  - ru
ID  - DM_2024_36_2_a3
ER  - 
%0 Journal Article
%A Yu. L. Pavlov
%T On the sizes of trees in a Galton–Watson forest with infinite variance in the critical case
%J Diskretnaya Matematika
%D 2024
%P 33-49
%V 36
%N 2
%U http://geodesic.mathdoc.fr/item/DM_2024_36_2_a3/
%G ru
%F DM_2024_36_2_a3
Yu. L. Pavlov. On the sizes of trees in a Galton–Watson forest with infinite variance in the critical case. Diskretnaya Matematika, Tome 36 (2024) no. 2, pp. 33-49. http://geodesic.mathdoc.fr/item/DM_2024_36_2_a3/

[1] Pavlov Yu. L., Cheplyukova I. A., “Ob'emy derevev sluchainogo lesa i konfiguratsionnye grafy”, Trudy MIAN, 316 (2022), 298–315 | DOI | Zbl

[2] Pavlov Yu. L., “Maksimalnoe derevo sluchainogo lesa v konfiguratsionnom grafe”, Matem. sb., 212:9 (2021), 146–163 | DOI | MR | Zbl

[3] Pavlov Yu. L., “O maksimalnom dereve lesa Galtona – Vatsona s beskonechnoi dispersiei raspredeleniya chisla potomkov”, Diskretnaya matematika, 35:2 (2023), 78–92 | DOI

[4] Hofstad R., Random Graphs and Complex Networks, v. 1, Cambridge Univ. Press, 2017, 328 pp. | Zbl

[5] Hofstad R., Random Graphs and Complex Networks, v. 2, Cambridge Univ. Press, 2024, 498 pp.

[6] Pavlov Yu. L., Sluchainye lesa, Karelskii nauchnyi tsentr RAN, Petrozavodsk, 1996, 259 pp.

[7] Kolchin V. F., Sluchainye otobrazheniya, Nauka, M., 1984, 207 pp.

[8] Feller V., Vvedenie v teoriyu veroyatnostei i ee primeneniya, v. 2, Mir, M., 1984, 752 pp. | MR

[9] Ibragimov I. A., Linnik Yu. V., Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965, 524 pp.

[10] Slack R. S., “A branching process with mean one and possibly infinite variance”, Z. Wahrscheinlichkeitstheorie verw. Geb., 9 (1968), 139–145 | DOI | MR | Zbl

[11] Khvorostyanskaya E. V., “O chisle derevev zadannogo ob'ema v lese Galtona – Vatsona v kriticheskom sluchae”, Teoriya veroyatnostei i ee primeneniya, 68:1 (2023), 75–92 | DOI | MR | Zbl

[12] Reittu H., Norros I., “On the power-law random graph model of massive data networks”, Perform. Eval., 55:1–2 (2004), 3–23 | DOI

[13] Newman M. E. J., “The structure and function of complex networks”, SIAM Review, 45:2 (2003), 107–256 | DOI | MR

[14] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1965, 296 pp. | MR