A note on stable point processes occurring in branching Brownian motion

We call a point process $Z$ on $\mathbb R$ \emph{exp-1-stable} if for every $\alpha,\beta\in\mathbb R$ with $e^\alpha+e^\beta=1$, $Z$ is equal in law to $T_\alpha Z+T_\beta Z'$, where $Z'$ is an independent copy of $Z$ and $T_x$ is the translation by $x$. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process $D$ on $\mathbb R$ such that $Z$ is equal in law to $\sum_{i=1}^\infty T_{\xi_i} D_i$, where $(\xi_i)_{i\ge1}$ are the atoms of a Poisson process of intensity $e^{-x}\,\mathrm d x$ on $\mathbb R$ and $(D_i)_{i\ge 1}$ are independent copies of $D$ and independent of $(\xi_i)_{i\ge1}$. In this note, we show how this decomposition follows from the classic \emph{LePage decomposition} of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb R$

Hollywood Loving

In this Essay, I highlight how nongovernmental entities establish political, moral, and sexual standards through visual media, which powerfully underscores and expresses human behavior. Through the Motion Picture Production Code (the “Hays Code”) and the Code of Practices for Television Broadcasters (the “TV Code”), Americans viewed entertainment as a pre-mediated, engineered world that existed outside of claims of censorship and propaganda. This Essay critically examines the role of film and television as persuasive and integral legal actors and it considers how these sectors operate to maintain, and sometimes challenge, racial order

The limiting process of NN-particle branching random walk with polynomial tails

We consider a system of $N$ particles on the real line that evolves through iteration of the following steps: 1) every particle splits into two, 2) each particle jumps according to a prescribed displacement distribution supported on the positive reals and 3) only the $N$ right-most particles are retained, the others being removed from the system. This system has been introduced in the physics literature as an example of a microscopic stochastic model describing the propagation of a front. Its behavior for large $N$ is now well understood -- both from a physical and mathematical viewpoint -- in the case where the displacement distribution admits exponential moments. Here, we consider the case of displacements with regularly varying tails, where the relevant space and time scales are markedly different. We characterize the behavior of the system for two distinct asymptotic regimes. First, we prove convergence in law of the rescaled positions of the particles on a time scale of order $\log N$ and give a construction of the limit based on the records of a space-time Poisson point process. Second, we determine the appropriate scaling when we let first the time horizon, then $N$ go to infinity.Comment: 17 pages, 1 figur