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Occupation time fluctuations of an age-dependent critical binary branching particle system

José Alfredo López-Mimbela, Antonio Murillo-Salas, José Hermenegildo Ramírez-González

Abstract

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.

Occupation time fluctuations of an age-dependent critical binary branching particle system

Abstract

We study the limit fluctuations of the rescaled occupation time process of a branching particle system in , where the particles are subject to symmetric -stable migration (), 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 -stable law with . In the latter case we show that, for dimensions , 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 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.
Paper Structure (14 sections, 12 theorems, 184 equations)

This paper contains 14 sections, 12 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Fluctuation limit theorems
  4. Weighted sub-fractional Brownian motion
  5. Laplace functional
  6. Proofs of main results
  7. Proof of theorems \ref{['main1']} and \ref{['main']}
  8. Tightness
  9. Space-time method: convergence to a Gaussian process
  10. Proof of Theorem \ref{['QgeneralTh']}.
  11. Proof of Lemma \ref{['QnotConvariance']}
  12. Proof of Theorem \ref{['limit-properties']} and Theorem \ref{['limit-Bm']}
  13. Proof of Theorem \ref{['limit-properties']}
  14. Proof of Theorem \ref{['limit-Bm']}

Key Result

Theorem 2.1

Let $F$ be an absolutely continuous lifetime distribution function satisfying (tail). Let $\alpha\gamma<d<\alpha(1+\gamma)$ and $H_T=T^{(2+\gamma-d/\alpha)/2}$. Then $\mathcal{J}_T\Rightarrow \mathcal{J}$ in $C([0, \Upsilon],{\cal S}^\prime(\mathbb{R}^d))$ as $T\rightarrow\infty$ for any $\Upsilon>0

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 10 more