English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study supercritical branching processes in which all particles evolveaccording to some general Markovian motion (which may possess absorbing states)and branch independently at a fixed constant rate. Under fairly naturalassumptions on the asymptotic distribution of the underlying motion, we firstshow using only probablistic tools that there is convergence in $L^2$ of theempirical measure (normalized by the mean number of particles) if and only ifan associated additive martingale is bounded in $L^2$. This is a significantimprovement over previous results, which were mainly restricted to$\lambda$-positive motions. We then investigate under which conditions thislimit is strictly positive on the event of non-extinction and show that thisoccurs whenever, on the latter event, particles do not accumulate on theboundary of the state space. In particular, this property also yields theconvergence of the real empirical measure (normalized by the number ofparticles). Moreover, building on previous results we prove that if the motionis $\lambda$-positive then these limits hold also almost surely whenever theDoob's $h$-transform of this motion admits a suitable Lyapunov functional.Finally, we illustrate our results for a variety of different motions: ergodicmotions without absorption, $\lambda$-positive systems either transient or withabsorption, and also certain non $\lambda$-positive systems such as theBrownian motion with negative drift killed at $0$. A strong law of largenumbers for the empirical measure in this last example was announced by Kestenin the late seventies, although a proof of it has remained undisclosed so far.Our results allow us to give a partial proof to Kesten's claim.

On the Kesten-Stigum theorem in $L^2$ beyond $\lambda$-positivity