A note on stable point processes occurring in branching Brownian motion

Introduction

Immediately after the article was published on the arXiv, the author was informed by Ilya Molchanov that the representation (1.1) could be obtained from a classic result known as the LePage decomposition of a stable point process, which holds true in much more general settings.

The purpose of this note is two-fold: First, we want to outline how the theory of stability in convex cones as developped by Davydov, Molchanov and Zuyev yields the above-mentioned LePage decomposition of stable point processes and with it the decomposition (1.1). This is the content of Section 2. Second, we give a succinct proof of the decomposition (1.1) for easy reference, a proof which uses more elementary methods than those of . Furthermore, we give the extension of (1.1) to random measures, which cannot be directly obtained through the results of (see Section 2). The statements of the results (Theorem 3.1 and Corollary 3.2) and their proofs are the content of Section 3.

In the remainder of this introduction, we outline the way exp-1-stable processes appear in branching Brownian motion (BBM). Define BBM as follows: Starting with one initial particle at the point $x$ of the real line, this particle performs Brownian motion until an exponentially distributed time of parameter $1/2$, at which it splits into two particles. Starting from the position of the split, both particles then repeat this process independently.

Once the convergence of the point process is established, one now readily sees that the limiting process is exp-1-stable : Take two BBMs and denote their derivative martingale limits by $W$ and $W^{\prime}$, respectively. The union of both processes is then again a BBM with derivative martingale limit $W^{\prime\prime}=W+W^{\prime}$. Applying the before-mentioned convergence result to both BBMs as well as to their union, we get that for almost every realisation of $W$ and $W^{\prime}$, $T_{\log(W+W^{\prime})}Z$ is equal in law to $T_{\log W}Z+T_{\log W^{\prime}}Z^{\prime}$, where $Z$ and $Z^{\prime}$ are iid and independent of $W$ and $W^{\prime}$. Since $W$ and $W^{\prime}$ can take any positive value (for example by varying the initial configurations), this yields the exp-1-stability of $Z$.

We emphasise that with this approach, one does not need to characterise the point process $Z$ directly, as it has been done before . This is helpful for models where such a direct characterisation would be complicated, for example for branching random walks .

Stability in convex cones

A succinct proof of the decomposition (1.1)

The following theorem and its corollary are precise statements of the decomposition (1.1) and form the main results of this paper.

Moreover, $\Delta$ can be chosen such that $\Delta(M(\mu)\neq 0)=0$, and as such, it is unique.

2 Infinitely divisible random measures

The main result about infinitely divisible random measures is the following (see , Theorem 6.1 or , Proposition 10.2.IX, however, note the error in the theorem statement of the latter reference: $F_{1}$ may be infinite as it is defined).

A random measure $Z$ with cumulant $K(f)$ is infinitely divisible if and only if

where $\lambda\in\mathcal{M}$ and $\Lambda$ is a measure on $\mathcal{M}^{*}$ satisfying

The probabilistic interpretation (, Lemma 6.5) of this fact is that $Z$ is the superposition of the non-random measure $\lambda$ and of the atoms of a Poisson process on $\mathcal{M}^{*}$ with intensity $\Lambda$. In the general framework of infinitely divisible distributions on semigroups used in the measures $\lambda$ and $\Lambda$ are called the Gaussian component and the Lévy measure, respectively. Fact 3.4 has the following analogue in the case of point processes (, Proposition 10.2.V), where the measure $\Lambda$ is also called the KLM measure. Note that the Gaussian component disappears.

3 Proof of Theorem 3.1

Furthermore, it is easy to show that exp-stability implies infinite divisibility. We then have the following lemma.

Let $\lambda,\Lambda$ be the measures corresponding to $Z$ by Fact 3.3.

The measures $T_{x}\lambda$, $T_{x}\Lambda$ are the measures corresponding to the infinitely divisible random measure $T_{x}Z$ by Fact 3.3. But by Corollary 3.5, the measures $e^{x}\lambda$ and $e^{x}\Lambda$ correspond to $T_{x}Z$, as well. Since these measures are unique, we have $T_{x}\lambda=e^{x}\lambda$ and $T_{x}\Lambda=e^{x}\Lambda$. The second statement follows immediately. For the first statement, note that $c_{1}=\lambda([0,1))<\infty$, since $[0,1)$ is a bounded set. It follows that

By monotonicity, the first integral is greater than or equal to $x\Lambda(\mu(I)>x)$ for every $x\in$, hence $\Lambda(\mu(I)>x)\leq C/x$ for some constant $0\leq C<\infty$. By the second statement, it follows that

4 Finiteness of the intensity

References

Acknowledgements

I thank two anonymous referees for having spotted several typographical errors in the manuscript and for having requested more details and explanations, which greatly benefitted the presentation. One referee pointed out the reference .