Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 12 (2007) > Paper 30 open journal systems 


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

Ben M Hambly, University of Oxford
Liza A Jones, University of Oxford


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.


Full text: PDF

Pages: 862-887

Published on: June 13, 2007
Bibliography
  1. 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. MR1326559
  2. Barbour, A. D.; Hall, Peter. On the rate of Poisson convergence. Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 3, 473--480. MR0755837
  3. Berry, M. V. The Bakerian lecture, 1987. Quantum chaology. Proc. Roy. Soc. London Ser. A 413 (1987), no. 1844, 183--198.MR0909277
  4. Berry, M. V. Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity 1 (1988), no. 3, 399--407. MR0955621
  5. Berry, M. V.; Keating, J. P. The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (1999), no. 2, 236--266 (electronic). MR1684543
  6. 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 MR1700749
  7. 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.MR0730191
  8. Costin, O. and Lebowitz, J. Gaussian fluctuations in random matrices}. Phys.Rev.Lett 75 (1995),69--72. Math Review number not available.
  9. Doob, J. L. Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp. MR0058896
  10. 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. MR0270403
  11. 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. MR2249630
  12. 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. MR2136367
  13. 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. MR0178516
  14. 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. MR1450904
  15. 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. Math Review number not available.
  16. Isham, Valerie; Westcott, Mark. A self-correcting point process. Stochastic Process. Appl. 8 (1978/79), no. 3, 335--347. MR0535308
  17. Johansson, Kurt. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001), no. 3, 683--705.MR1810949
  18. Johansson, Kurt. Determinantal processes with number variance saturation. Comm. Math. Phys. 252 (2004), no. 1-3, 111--148. MR2103906
  19. 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). MR2249664
  20. 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 MR1876169
  21. Luo, W.; Sarnak, P. Number variance for arithmetic hyperbolic surfaces. Comm. Math. Phys. 161 (1994), no. 2, 419--432.MR1266491
  22. Mandelbrot, Benoit B.; Van Ness, John W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 1968 422--437. MR0242239
  23. 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. MR2129906
  24. 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. MR1280932
  25. 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. MR1739520
  26. Shanbhag, D. N.; Sreehari, M. On certain self-decomposable distributions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), no. 3, 217--222.MR0436267
  27. Soshnikov, Alexander. Level spacings distribution for large random matrices: Gaussian fluctuations. Ann. of Math. (2) 148 (1998), no. 2, 573--617. MR1668559
  28. 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. MR1799012
  29. 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. MR1788476
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489