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 $\mathbb R$

Boundary Crossing Probabilities for General Exponential Families

We consider parametric exponential families of dimension $K$ on the real line. We study a variant of \textit{boundary crossing probabilities} coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension $K$. Formally, our result is a concentration inequality that bounds the probability that $\mathcal{B}^\psi(\hat \theta_n,\theta^\star)\geq f(t/n)/n$, where $\theta^\star$ is the parameter of an unknown target distribution, $\hat \theta_n$ is the empirical parameter estimate built from $n$ observations, $\psi$ is the log-partition function of the exponential family and $\mathcal{B}^\psi$ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function $f$ is logarithmic, as it is enables to analyze the regret of the state-of-the-art \KLUCB\ and \KLUCBp\ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when $K=1$, while we provide results for arbitrary finite dimension $K$, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T.L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits

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