arXiv subjects

Xinxin Chen

Publications and source records attributed to Xinxin Chen.

40 records · Page 3Linked to original sources

arXiv2014

Increasing paths on N-ary trees

Consider a rooted $N$-ary tree. To every vertex of this tree, we attach an i.i.d. continuous random variable. A vertex is called accessible if along its ancestral line, the attached random variables are increasing. We keep accessible vertices and kill all the others. For any positive constant $α$, we describe the asymptotic behaviors of the population at the $αN$-th generation as $N$ goes to infinity. We also study the criticality of the survival probability at the $(eN-\frac{3}{2}\log N)$-th generation in this paper.

math.PR↗

arXiv2014

A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk

We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and sufficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.

math.PR↗

arXiv2013

Scaling limit of the path leading to the leftmost particle in a branching random walk

We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the $n$-th generation behaves asymptotically like $\frac{3}{2}\ln n$, provided the non-extinction of the system. The main goal of this paper, is to prove that the path from the root to the leftmost particle, after a suitable normalizatoin, converges weakly to a Brownian excursion in $D([0,1],\r)$.

math.PR↗

arXiv2013

Waiting times for particles in a branching Brownian motion to reach the rightmost position

It has been proved by Lalley and Sellke [13] that every particle born in a branching Brownian motion has a descendant reaching the rightmost position at some future time. The main goal of the present paper is to estimate asymptotically as s goes to infinity, the first time that every particle alive at the time s has a descendant reaching the rightmost position.

math.PR↗