Geodesic
Limit theorems for the number of trees of a given size in a random forest
Sbornik. Mathematics, Tome 32 (1977) no. 3, pp. 335-345 
Yu. L. Pavlov. Limit theorems for the number of trees of a given size in a random forest. Sbornik. Mathematics, Tome 32 (1977) no. 3, pp. 335-345. http://geodesic.mathdoc.fr/item/SM_1977_32_3_a4/
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     title = {Limit theorems for the number of trees of a~given size in a~random forest},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author considers the set of all forests consisting of $N$ rooted trees and containing $n$ nonroot vertices; the root vertices are numbered from 1 to $N$, and the nonroot from 1 to $n$. A uniform probability distribution is introduced on this set. Let $\mu_r(n,N)$ denote a random variable equal to the number of trees of a random forest containing exactly $r$ nonroot vertices. Results are obtained yielding a complete description of the limit behavior of the variables $\mu_r(n,N)$ for all values of $r$ for various ways of letting $n$ and $N$ approach infinity. It is shown that these results can be used for studying random mappings. Bibliography: 9 titles.

[1] V. E. Stepanov, “O raspredelenii chisla vershin v sloyakh sluchainogo dereva”, Teoriya veroyatn., XIV:1 (1969), 64–77 | MR

[2] V. F. Kolchin, “Odin klass predelnykh teorem dlya uslovnykh raspredelenii”, Litovskii matem. sb., VIII:1 (1968), 53–63 | MR

[3] A. I. Markushevich, Teoriya analiticheskikh funktsii, Gostekhizdat, Moskva, 1950

[4] V. F. Kolchin, B. A. Sevastyanov, V. P. Chistyakov, Sluchainye razmescheniya, izd-vo «Nauka», Moskva, 1976 | MR

[5] V. E. Stepanov, “Predelnye raspredeleniya nekotorykh kharakteristik sluchainykh otobrazhenii”, Teoriya veroyatn., XIV:4 (1969), 639–653 | MR

[6] Yu. V. Prokhorov, Yu. A. Rozanov, Teoriya veroyatnostei. Osnovnye ponyatiya. Predelnye teoremy. Sluchainye protsessy, izd-vo «Nauka», Moskva, 1973 | MR

[7] B. V. Gnedenko, A. N. Kolmogorov, Predelnye raspredeleniya dlya summ nezavisimykh sluchainykh velichin, Gostekhizdat, Moskva–Leningrad, 1949

[8] S. G. Tkachuk, “Lokalnye predelnye teoremy, uchityvayuschie bolshie ukloneniya v sluchae predelnykh ustoichivykh zakonov”, Izv. AN UzSSR, seriya fiz-matem. nauk, 2 (1973), 30–33 | Zbl

[9] B. Harris, “Probability distributions related to random mappings”, Ann. Math. Statist., 31:4 (1960), 1045–1062 | DOI | MR | Zbl