Revisiting Offspring Maxima in Branching Processes

On the maximal offspring in a subcritical branching process

We consider a subcritical Galton--Watson tree conditioned on having $n$ vertices with outdegree in a fixed set $Ω$. Under mild regularity assumptions we prove various limits related to the maximal offspring of a vertex as $n$ tends to infinity.

一类具有稳定后代分布的临界分支Lévy过程的最大位移研究

On the maximal displacement of some critical branching L{é}vy processes with stable offspring distribution

该研究针对后代分布属稳定随机变量吸引域的临界分支Lévy过程，在底层过程非中心化且有至少二阶矩、分布具正则变化尾两种情形下，分析粒子最大位置的尾概率，补充了既有研究的相关结果。

On largest offsprings in a critical branching process with finite variance

We continue our study of the distribution of the maximal number $X^{\ast}_k$ of offsprings amongst all individuals in a critical Galton-Watson process started with $k$ ancestors, treating the case when the reproduction law has a regularly varying tail $\bar F$ with index $-α$ for $α>2$ (and hence finite variance). We show that $X^{\ast}_k$ suitably normalized converges in distribution to a Frechet law with shape parameter $α/2$; this contrasts sharply with the case $1<α<2$ when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in the dec…

Revisiting John Lamperti's maximal branching process

Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.

临界与次临界分支稳定过程中最大位移的尾部概率

Tail probability of maximal displacement in critical and subcritical branching stable processes

本文研究临界与次临界分支α-稳定过程最大位移的尾部概率，针对正跳跃、谱负及临界情形分别给出精确渐近行为。

具有给定度的多部随机图：局部极限、重新审视巨分支、距离

Multipartite random graphs with given degrees: local limit, revisiting the giant, distances

研究具有给定度序列的多部随机图，证明其局部极限是多类型分支过程，去除前人假设，给出提取巨分支通用框架及多类型分支过程新生存准则。