[Paper Review] A characterisation of superposable random measures

Pascal Maillard|arXiv (Cornell University)|Feb 9, 2011

Stochastic processes and statistical mechanics4 references7 citations

TL;DR

This paper characterizes exp-1-stable random measures on ℝ using a generalization of the LePage decomposition, proving that any such measure admits a superposition representation as a random sum of i.i.d. copies of a base random measure, indexed by a Poisson process with intensity e^{-x}dx. The key contribution is a concise, general proof of this decomposition for random measures, extending prior results in branching processes and stable processes.

ABSTRACT

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 $\mathbb R$.

Motivation & Objective

• To provide a general characterization of exp-1-stable random measures on ℝ beyond specific stochastic processes.
• To unify and generalize existing results on point process decompositions in branching Brownian motion and branching random walk.
• To establish a rigorous and concise proof of the superposition decomposition using classical stochastic tools.
• To extend the LePage decomposition framework to the broader class of random measures, not just point processes.

Proposed method

• The paper employs the classical LePage decomposition for union-stable random measures as a foundational tool.
• It defines exp-1-stability via a scaling-invariance condition involving translations and independent copies.
• The authors use a Poisson point process with intensity e^{-x}dx on ℝ to index the locations of i.i.d. copies of a base random measure.
• The decomposition is derived by analyzing the stability condition and applying moment generating function techniques in the context of random measures.
• The proof leverages the independence and identical distribution of the components to verify the law of the sum matches the original measure.
• The argument is generalized from point processes to general random measures, maintaining the same structural form.

Experimental results

Research questions

• RQ1 How can the superposition decomposition of exp-1-stable point processes be generalized to random measures on ℝ?
• RQ2 What is the minimal set of conditions under which a random measure admits a LePage-type decomposition?
• RQ3 Can the decomposition be derived directly from the stability condition without relying on specific process dynamics?
• RQ4 What role does the Poisson process with intensity e^{-x}dx play in characterizing the structure of such measures?
• RQ5 Is the representation as a sum of i.i.d. components with Poisson-distributed locations unique and necessary?

Key findings

• Any exp-1-stable random measure on ℝ admits a representation as a superposition of i.i.d. copies of a base random measure, indexed by a Poisson process with intensity e^{-x}dx.
• The decomposition is a direct consequence of the LePage decomposition framework, extended to general random measures.
• The proof establishes the equivalence in law between the original measure and the constructed superposition without requiring assumptions on the underlying stochastic process.
• The result generalizes prior findings in branching processes, such as those for extremal particles in branching Brownian motion.
• The method provides a short, self-contained proof applicable to both point processes and general random measures.
• The structure reveals a deep connection between stability under translation and Poisson-driven random measures.