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 $D([0,1],\r)$

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 $\alpha$, we describe the asymptotic behaviors of the population at the $\alpha N$-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

Does Taking One Step Back Get You Two Steps Forward? Grade Retention and School Performance in Rural China

Abstract: Despite the rise in grade retention in China recently, little work has been done to understand the impact of grade retention on the educational performance of students in China. This paper seeks to redress this shortcoming and examines this impact on 1649 students in 36 elementary schools in Shaanxi province. With a dataset that was collected from a survey designed specifically to capture school performance of students before and after they were retained, we use Difference-in-Difference, Propensity Score Matching and Difference-in-Difference Matching approaches to analyze the effect of grade retention on school performance. Although the descriptive analysis shows that grade retention helps to improve the scores of the students that were retained, somewhat surprisingly, the results from the multivariate analysis consistently show that there is no significant positive effect of grade retention on school performance of the students. In fact, in some cases (e.g., for the students who repeat grade 2), grade retention is shown to hurt school performance.Educational economics; Human capital