Electronic Journal of Probability

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

Ben Hambly and Liza Jones

Full-text: Open access

Abstract

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct explicit 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 associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 30, 862-887.

Dates
Accepted: 13 June 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818501

Digital Object Identifier
doi:10.1214/EJP.v12-419

Mathematical Reviews number (MathSciNet)
MR2318413

Zentralblatt MATH identifier
1127.60046

Subjects
Primary: 60G52: Stable processes 60G15: Gaussian processes
Secondary: 60F17: Functional limit theorems; invariance principles 15A52

Keywords
Number variance symmetric alpha- stable processes controlled variability Gaussian fluctuations functional limits long memory Gaussian processes fractional Brownian motion

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Hambly, Ben; Jones, Liza. Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes. Electron. J. Probab. 12 (2007), paper no. 30, 862--887. doi:10.1214/EJP.v12-419. https://projecteuclid.org/euclid.ejp/1464818501


Export citation

References

  • Aurich, R.; Steiner, F. Periodic-orbit theory of the number variance $\Sigma^2(L)$ of strongly chaotic systems. Phys. D 82 (1995), no. 3, 266–287.
  • Barbour, A. D.; Hall, Peter. On the rate of Poisson convergence. Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 3, 473–480.
  • Berry, M. V. The Bakerian lecture, 1987. Quantum chaology. Proc. Roy. Soc. London Ser. A 413 (1987), no. 1844, 183–198.
  • Berry, M. V. Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity 1 (1988), no. 3, 399–407.
  • Berry, M. V.; Keating, J. P. The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (1999), no. 2, 236–266 (electronic).
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Bohigas, O.; Giannoni, M.-J.; Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52 (1984), no. 1, 1–4.
  • Costin, O. and Lebowitz, J. Gaussian fluctuations in random matrices}. Phys.Rev.Lett 75 (1995),69–72.
  • Doob, J. L. Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Goldstein, S.; Lebowitz, J. L.; Speer, E. R. Large deviations for a point process of bounded variability. Markov Process. Related Fields 12 (2006), no. 2, 235–256.
  • Gorostiza, Luis G.; Navarro, Reyla; Rodrigues, Eliane R. Some long-range dependence processes arising from fluctuations of particle systems. Acta Appl. Math. 86 (2005), no. 3, 285–308.
  • Lewis, T.; Govier, L. J. Some properties of counts of events for certain types of point process. J. Roy. Statist. Soc. Ser. B 26 1964 325–337.
  • Guhr, Thomas; Mller-Groeling, Axel. Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales. Quantum problems in condensed matter physics. J. Math. Phys. 38 (1997), no. 4, 1870–1887.
  • Guhr, T. and Papenbrock, T. Spectral correlations in the crossover transition from a superposition of harmonic oscillators to the Gaussian unitary ensemble. Phys.Rev.E. 59 (1999), no. 1, 330–336.
  • Isham, Valerie; Westcott, Mark. A self-correcting point process. Stochastic Process. Appl. 8 (1978/79), no. 3, 335–347.
  • Johansson, Kurt. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001), no. 3, 683–705.
  • Johansson, Kurt. Determinantal processes with number variance saturation. Comm. Math. Phys. 252 (2004), no. 1-3, 111–148.
  • Jones, Liza; O'Connell, Neil. Weyl chambers, symmetric spaces and number variance saturation. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 91–118 (electronic).
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Luo, W.; Sarnak, P. Number variance for arithmetic hyperbolic surfaces. Comm. Math. Phys. 161 (1994), no. 2, 419–432.
  • Mandelbrot, Benoit B.; Van Ness, John W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 1968 422–437.
  • Mehta, Madan Lal. Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp. ISBN: 0-12-088409-7.
  • Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0.
  • Sato, Ken-iti. Lvy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4.
  • Shanbhag, D. N.; Sreehari, M. On certain self-decomposable distributions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), no. 3, 217–222.
  • Soshnikov, Alexander. Level spacings distribution for large random matrices: Gaussian fluctuations. Ann. of Math. (2) 148 (1998), no. 2, 573–617.
  • Soshnikov, A. Determinantal random point fields. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160; translation in Russian Math. Surveys 55 (2000), no. 5, 923–975.
  • Soshnikov, Alexander B. Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Statist. Phys. 100 (2000), no. 3-4, 491–522.