On the small deviation problem for some iterated processes

Frank Aurzada (Technische Universit├Ąt Berlin)
Mikhail Lifshits (St. Petersburg State University)


We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Electron. J. Probab. 11 (2006), no. 18, 434--459.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1992-2010

Publication Date: September 28, 2009

DOI: 10.1214/EJP.v14-689


  1. Allouba, Hassan; Zheng, Weian. Brownian-time processes: the PDE connection and the half-derivative generator. Ann. Probab. 29 (2001), no. 4, 1780--1795. MR1880242 (2002j:60118)
  2. Anderson, Theodore W. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, (1955). 170--176. MR0069229 (16,1005a)
  3. Aurzada, Frank. Small deviations for stable processes via compactness properties of the parameter set. Statist. Probab. Lett. 78 (2008), no. 6, 577--581. MR2409520 (2009h:60083)
  4. Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564 (98e:60117)
  5. Bingham, Nick H.; Goldie, Charles M.; Teugels, Jozef L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1 MR1015093 (90i:26003)
  6. Blackburn, Robert. Large deviations of local times of Lévy processes. J. Theoret. Probab. 13 (2000), no. 3, 825--842. MR1785531 (2001h:60084)
  7. Borovkov, A. A.; Mogulʹskiĭ, A. A. On probabilities of small deviations for stochastic processes [translation of Trudy Inst. Mat. (Novosibirsk) 13 (1989), Asimptot. Analiz Raspred. Sluch. Protsess., 147--168; MR1037254 (91e:60089)]. Siberian Advances in Mathematics. Siberian Adv. Math. 1 (1991), no. 1, 39--63. MR1100316
  8. Burdzy, Krzysztof. Some path properties of iterated Brownian motion. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 67--87, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993. MR1278077 (95c:60075)
  9. Burdzy, Krzysztof. Variation of iterated Brownian motion. Measure-valued processes, stochastic partial differential equations, and interacting systems (Montreal, PQ, 1992), 35--53, CRM Proc. Lecture Notes, 5, Amer. Math. Soc., Providence, RI, 1994. MR1278281 (95h:60123)
  10. Embrechts, Paul; Maejima, Makoto. Selfsimilar processes. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2002. xii+111 pp. ISBN: 0-691-09627-9 MR1920153 (2004c:60003)
  11. Hu, Yueyun; Pierre-Loti-Viaud, Daniel; Shi, Zhan. Laws of the iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 8 (1995), no. 2, 303--319. MR1325853 (96b:60073)
  12. Khoshnevisan, Davar; Lewis, Thomas M. Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 3, 349--359. MR1387394 (97k:60218)
  13. Khoshnevisan, Davar; Xiao, Yimin. Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33 (2005), no. 3, 841--878. MR2135306 (2006d:60078)
  14. Lacey, Michael. Large deviations for the maximum local time of stable Lévy processes. Ann. Probab. 18 (1990), no. 4, 1669--1675. MR1071817 (91h:60085)
  15. Li, Wenbo V. A Gaussian correlation inequality and its applications to small ball probabilities. Electron. Comm. Probab. 4 (1999), 111--118 (electronic). MR1741737 (2001j:60074)
  16. Li, Wenbo V.; Shao, Qi-Man. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533--597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001. MR1861734
  17. Lifshits, Mikhail; Simon, Thomas. Small deviations for fractional stable processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 725--752. MR2144231 (2006d:60081)
  18. Lifshits, Mikhail. Asymptotic behavior of small ball probabilities. Probab. Theory and Math. Statist. Proc. VII International Vilnius Conference, 1999, pp. 453--468. Math. Review number not available.
  19. Lifshits, Mikhail. Bibliography compilation on small deviation probabilities, available from http://www.proba.jussieu.fr/pageperso/smalldev/biblio.html, 2009. Math. Review number not available.
  20. Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231 (86e:60089)
  21. Linde, Werner; Shi, Zhan. Evaluating the small deviation probabilities for subordinated Lévy processes. Stochastic Process. Appl. 113 (2004), no. 2, 273--287. MR2087961 (2005h:60139)
  22. Linde, Werner; Zipfel, Pia. Small deviation of subordinated processes over compact sets. Probab. Math. Statist. 28 (2008), no. 2, 281--304. Math. Review number not yet available.
  23. Mogulʹskiĭ, A. A. Small deviations in the space of trajectories. (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755--765. MR0370701 (51 #6927)
  24. Nane, Erkan. Laws of the iterated logarithm for $alpha$-time Brownian motion. Electron. J. Probab. 11 (2006), no. 18, 434--459 (electronic). MR2223043 (2007c:60087)
  25. Nourdin, Ivan; Peccati, Giovanni. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2008), no. 43, 1229--1256. MR2430706 (2009h:60052)
  26. Pruitt, William E. The Hausdorff dimension of the range of a process with stationary independent increments. J. Math. Mech. 19 1969/1970 371--378. MR0247673 (40 #936)
  27. Samorodnitsky, Gennady. Lower tails of self-similar stable processes. Bernoulli 4 (1998), no. 1, 127--142. MR1611887 (99e:60105)
  28. 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 (95f:60024)
  29. Shi, Zhan. Lower limits of iterated Wiener processes. Statist. Probab. Lett. 23 (1995), no. 3, 259--270. MR1340161 (96m:60077)
  30. Taylor, S. James. Sample path properties of a transient stable process. J. Math. Mech. 16 1967 1229--1246. MR0208684 (34 #8493)
  31. Vervaat, Wim. Sample path properties of self-similar processes with stationary increments. Ann. Probab. 13 (1985), no. 1, 1--27. MR0770625 (86c:60063)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.