A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process - Archive ouverte HAL Access content directly
Journal Articles Bernoulli Year : 2023

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

Bastien Mallein
Quan Shi

#### Abstract

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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

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

### Identifiers

• HAL Id : hal-03251327 , version 2
• ARXIV :
• DOI :

### Cite

Bastien Mallein, Quan Shi. A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process. Bernoulli, 2023, 29 (1), pp.597-624. ⟨10.3150/22-BEJ1470⟩. ⟨hal-03251327v2⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

189 View