A new approximation of the Height process of a CSBP

Approximation of the Height process of a CSBP with interaction

In this work, we first show that the properly rescaled height process of the genealogical tree of a continuous time branching process converges to the height process of the genealogy of a (possibly discontinuous) continuous state branching process. We then prove the same type of result for generalized branching processes with interaction.

Convergence to a Continuous State Branching Process with jumps and Height Process

In this work, we study asymptotics of the genealogy of Galton-Watson processes. Thus we consider a offspring distribution such that the rescaled Galton-Watson processes converges to a continuous state branching process (CSBP) with jumps. After we show that the rescaled height (or exploration) process of the corresponding Galton-Watson family tree, converges in a functional sense, to the continuous height process that Le Gall and Le Jan introduced in 1998 on their paper "branching processes in Lévy processes : The exploration process".

A limit theorem for the contour process of conditioned Galton-Watson trees

In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $α\in (1, 2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced \cite{LGLJ1, LGLJ1}. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one…

Proof(s) of the Lamperti representation of Continuous-State Branching Processes

This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive Lévy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.

大小条件 Bienaymé 树中的稀有子树模式：泊松近似与解聚

Rare subtree patterns in size-conditioned Bienaymé trees: Poisson approximation and declumping

研究临界 Bienaymé-Galton-Watson 树中稀有子树模式，结合多种方法克服大小条件依赖性，建立泊松近似，应用于最大叶高、完全$r$元树高度等，给出不同情况下的渐近结果及行为特点。

A study of large fringe and non-fringe subtrees in conditional Galton-Watson trees

We study the conditions for families of subtrees to exist with high probability (whp) in a Galton-Walton tree of size $n$. We first give a Poisson approximation of fringe subtree counts, which yields the height of the maximal complete $r$-ary fringe subtree. Then we determine the maximal $K_n$ such that every tree of size at most $K_n$ appears as fringe subtree whp. Finally, we study non-fringe subtree counts and determine the height of the maximal complete $r$-ary non-fringe subtree.