A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année :

## A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

(1) , (2)
1
2
Bastien Mallein
Quan Shi

#### Résumé

A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.

#### Domaines

Mathématiques [math] Probabilités [math.PR]

### Dates et versions

hal-03251327 , version 1 (26-01-2022)
hal-03251327 , version 2 (17-11-2022)

### Identifiants

• HAL Id : hal-03251327 , version 1
• ARXIV :

### Citer

Bastien Mallein, Quan Shi. A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process. 2022. ⟨hal-03251327v1⟩

### Exporter

BibTeX TEI Dublin Core DC Terms EndNote Datacite
47 Consultations
16 Téléchargements

### Partager

Gmail Facebook Twitter LinkedIn More