Abstract
It is shown that the conditional distributions of a number of characteristics of a branching process μ(t), μ(0)=m, under the condition that the number of total progeny μm in this process is equal to n, coincide with the distributions of the corresponding characteristics of a generalized scheme of arrangement of particles in cells. In the case where the number of offsprings of a particle has the Poisson distribution, the characteristics of the branching process μ(t), μ(0)=1, under the condition that ν1=n+1, coincide with the characteristics of a random tree. By using these connections we obtain in this article a series of limit theorems as n→∞ for characteristics of random trees and branching processes under the conditions that νm=n.
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Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 691–705, May, 1977.
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Kolchin, V.F. Branching processes, random trees, and a generalized scheme of arrangements of particles. Mathematical Notes of the Academy of Sciences of the USSR 21, 386–394 (1977). https://doi.org/10.1007/BF01788236
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DOI: https://doi.org/10.1007/BF01788236