This article is cited in 2 papers

On the sizes of trees in a Galton–Watson forest with infinite variance in the critical case

Yu. L. Pavlov

Karelian Research Centre of the Russian Academy of Sciences

Abstract: 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.

UDC: 519.212.2+519.179.4

Received: 28.01.2024

DOI: 10.4213/dm1813