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 Electronic Communications in Probability > Vol. 12 (2007) > Paper 17 open journal systems 


Some Extensions of Fractional Brownian Motion and Sub-Fractional Brownian Motion Related to Particle Systems

Tomasz Bojdecki, Institute of Mathematics, University of Warsaw
Luis G Gorostiza, Centro de Investigacion y de Estudios Avanzados, Mexico
Anna Talarczyk, Institute of Mathematics, University of Warsaw


Abstract
In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance
0min(s,t) ua [(t-u)b+(s-u)b]du,
parameters a > -1, -1 < b ≤ 1, |b| ≤ 1 + a, corresponds to fractional Brownian motion for a = 0, -1 < b < 1. The second one, with covariance
(2-h)(sh + th - (1/2)[(s+t)h + |s-t|h]),
parameter 0 < h ≤ 4, corresponds to sub-fractional Brownian motion for 0 < h < 2. The third one, with covariance
-(s2log s + t2log t -(1/2)[(s+t)2 log (s+t) +(s-t)2 log |s-t|]),
is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.


Full text: PDF | PostScript

Pages: 161-172

Published on: May 16, 2007


Bibliography
  1. Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A. Fractional Brownian density process and its self-intersection local time of order $k$. J. Theoret. Probab. 17 (2004), no. 3, 717--739. MR2091558(2005j:60076)
  2. Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Sub-fractional Brownian motion and its relation to occupation times. Statist. Probab. Lett. 69 (2004), no. 4, 405--419. MR2091760 (2005k:60124)
  3. Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A. Limit theorems for occupation time fluctuations of branching systems. I. Long-range dependence. Stochastic Process. Appl. 116 (2006), no. 1, 1--18. MR2186101 (2007b:60083)
  4. Bojdecki, T.; Gorostiza, L.G.; Talarczyk, A. A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 (2007), no. 2, 500--527.
  5. Bojdecki, T.; Gorostiza, L.G.; Talarczyk, A. Occupation time limits of inhomogeneous Poisson systems of independent particles, Stoch. Proc. Appl. (to appear).
  6. Bojdecki, T.; Gorostiza, L.G.; Talarczyk, A. Self-similar stable processes arising from high density limits of occupation times of particle systems, in preparation.
  7. Theory and applications of long-range dependence. Edited by Paul Doukhan, George Oppenheim and Murad S. Taqqu. Birkhäuser Boston, Inc., Boston, MA, 2003. xii+719 pp. ISBN: 0-8176-4168-8 MR1956041 (2003h:60004)
  8. Dzhaparidze, Kacha; van Zanten, Harry. A series expansion of fractional Brownian motion. Probab. Theory Related Fields 130 (2004), no. 1, 39--55. MR2092872 (2005i:60065)
  9. Fernique, Xavier. Intégrabilité des vecteurs gaussiens. (French) C. R. Acad. Sci. Paris Sér. A-B 270 1970 A1698--A1699. MR0266263 (42 #1170)
  10. Gorostiza, Luis G.; Wakolbinger, Anton. Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 (1991), no. 1, 266--288. MR1085336 (91k:60089)
  11. Houdré, Christian; Villa, José. An example of infinite dimensional quasi-helix. Stochastic models (Mexico City, 2002), 195--201, Contemp. Math., 336, Amer. Math. Soc., Providence, RI, 2003. MR2037165 (2004m:60077)
  12. Iscoe, I. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71 (1986), no. 1, 85--116. MR0814663 (87c:60070)
  13. 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 (2002m:60002)
  14. Russo, Francesco; Tudor, Ciprian A. On bifractional Brownian motion. Stochastic Process. Appl. 116 (2006), no. 5, 830--856. MR2218338(2006k:60070)
  15. Talarczyk, A. A functional ergodic theorem for the occupation time process of a branching system, (submitted).
  16. Taqqu, M.S. Self-similarity and long-range dependence, 10th Brazilian School of Probability and 2006 Annual Meeting of the Institute of Mathematical Statistics, IMPA, Rio de Janeiro, August, 2006.
  17. Tudor, Constantin. Some properties of mild solutions of delay stochastic evolution equations. Stochastics 17 (1986), no. 1-2, 1--18. MR0878551 (89k:60084)
  18. Tudor, C. Inner product spaces of integrands associated to sub-fractional Brownian motion, preprint.
  19. Tudor C.A.; Xiao, Y. Sample path properties of bifractional Brownian motion, Math. ArXiv PR/0606753.
















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Electronic Communications in Probability. ISSN: 1083-589X