Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.11540 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909097072263168 |
|---|---|
| author | López-Mimbela, José Alfredo Murillo-Salas, Antonio Ramírez-González, José Hermenegildo |
| author_facet | López-Mimbela, José Alfredo Murillo-Salas, Antonio Ramírez-González, José Hermenegildo |
| contents | We study the limit fluctuations of the rescaled occupation time process of a branching particle system in $\mathbb{R}^d$, where the particles are subject to symmetric $α$-stable migration ($0<α\leq2$), critical binary branching, and general non-lattice lifetime distribution. We focus on two different regimes: lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.e. distributions belonging to the normal domain of attraction of a $γ$-stable law with $γ\in(0,1)$.
In the latter case we show that, for dimensions $αγ<d<α(1+γ)$, the rescaled occupation time fluctuations converge weakly to a centered Gaussian process whose covariance function is explicitly calculated, and we call it {\em weighted sub-fractional Brownian motion.} Moreover, in the case of lifetimes with finite mean, we show that for $α<d<2α$ the fluctuation limit turns out to be the same as in the case of exponentially distributed lifetimes studied by Bojdecki et al. [7,8,9]. We also investigate the maximal parameter range allowing existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, such as path continuity, long-range dependence, self-similarity and the lack of Markov property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Occupation time fluctuations of an age-dependent critical binary branching particle system López-Mimbela, José Alfredo Murillo-Salas, Antonio Ramírez-González, José Hermenegildo Probability 60J80, 60E10 We study the limit fluctuations of the rescaled occupation time process of a branching particle system in $\mathbb{R}^d$, where the particles are subject to symmetric $α$-stable migration ($0<α\leq2$), critical binary branching, and general non-lattice lifetime distribution. We focus on two different regimes: lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.e. distributions belonging to the normal domain of attraction of a $γ$-stable law with $γ\in(0,1)$. In the latter case we show that, for dimensions $αγ<d<α(1+γ)$, the rescaled occupation time fluctuations converge weakly to a centered Gaussian process whose covariance function is explicitly calculated, and we call it {\em weighted sub-fractional Brownian motion.} Moreover, in the case of lifetimes with finite mean, we show that for $α<d<2α$ the fluctuation limit turns out to be the same as in the case of exponentially distributed lifetimes studied by Bojdecki et al. [7,8,9]. We also investigate the maximal parameter range allowing existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, such as path continuity, long-range dependence, self-similarity and the lack of Markov property. |
| title | Occupation time fluctuations of an age-dependent critical binary branching particle system |
| topic | Probability 60J80, 60E10 |
| url | https://arxiv.org/abs/2301.11540 |