Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric -stable processes.

@article{Hambly2006NumberVF,
  title={Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric -stable processes.},
  author={Ben M. Hambly and L. A. Jones},
  journal={arXiv: Probability},
  year={2006}
}
Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric -stable processes are used to construct new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the… 
Particle systems with quasi-homogeneous initial states and their occupation time fluctuations
We consider particle systems in $R$ with initial configurations belonging to a class of measures that obey a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures
Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses
The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0 d)$. In some cases $H_T\equiv 1$ and

References

SHOWING 1-10 OF 37 REFERENCES
Some Long-Range Dependence Processes Arising from Fluctuations of Particle Systems
Abstract Several long-range dependence, self-similar Gaussian processes arise from asymptotics of some classes of spatially distributed particle systems and superprocesses. The simplest examples are
Determinantal Processes with Number Variance Saturation
Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process
Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance
Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable
Large Deviations for a Point Process of Bounded Variability
We consider a one-dimensional translation invariant point process of density one with uniformly bounded variance of the number NI of particles in any interval I . Despite this suppression of
Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields
We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem,
Level spacings distribution for large random matrices: Gaussian fluctuations
We study the level-spacings distribution for eigenvalues of large N X N matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1.
Foundations of modern probability
* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit
Semiclassical formula for the number variance of the Riemann zeros
By pretending that the imaginery parts Em of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it
Lévy processes and infinitely divisible distributions
Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5.
...
...