Electronic Journal of Probability

Laws of the iterated logarithm for $\alpha$-time Brownian motion

Erkan Nane

Full-text: Open access

Abstract

We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0 < \alpha \leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\alpha$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in Hu (1995) for iterated Brownian motion. When $\alpha =1$ it takes the following form $$ \liminf_{T\to\infty}\ T^{-1/2}(\log\log T) \sup_{0\leq t\leq T}|Z_{t}|=\pi^{2}\sqrt{\lambda_{1}} \quad a.s. $$ where $\lambda_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}(t)=|{x: Z(s)=x \text{ for some } s\leq t}|$ for these processes for $1 <\alpha <2$. We prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $ such that $$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2\alpha}\log \log t}= c_{R} \quad a.s. $$ $$ \liminf_{t\to\infty} \frac{\sup_{x\in {R}}L^{*}(x,t)}{(t/\log \log t)^{1-1/2\alpha}}= c_{L} \quad a.s. $$

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 18, 434-459.

Dates
Accepted: 19 June 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730553

Digital Object Identifier
doi:10.1214/EJP.v11-327

Mathematical Reviews number (MathSciNet)
MR2223043

Zentralblatt MATH identifier
1121.60085

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60K99: None of the above, but in this section

Keywords
Brownian motion symmetric $alpha$-stable process $alpha$-time Brownian motion local time Chung's law Kesten's law

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

Citation

Nane, Erkan. Laws of the iterated logarithm for $\alpha$-time Brownian motion. Electron. J. Probab. 11 (2006), paper no. 18, 434--459. doi:10.1214/EJP.v11-327. https://projecteuclid.org/euclid.ejp/1464730553


Export citation

References

  • Allouba, Hassan; Zheng, Weian. Brownian-time processes: the PDE connection and the half-derivative Ann. Probab. 29 (2001), no. 4, 1780–1795.
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
  • Burdzy, Krzysztof. Some path properties of iterated Brownian motion. 67–87, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993.
  • Burdzy, Krzysztof; Khoshnevisan, Davar. The level sets of iterated Brownian motion. 231–236, Lecture Notes in Math., 1613, Springer, Berlin, 1995.
  • Chen, Xia; Li, Wenbo V. Small deviation estimates for some additive processes. 225–238, Progr. Probab., 55, Birkhäuser, Basel, 2003.
  • Yin, Chuan Cun; Lü, Yu Hua. A Chung-type law of the iterated logarithm for iterated Brownian (Chinese) Acta Math. Sinica 43 (2000), no. 1, 99–106.
  • Chung, Kai Lai. On the maximum partial sums of sequences of independent random Trans. Amer. Math. Soc. 64, (1948). 205–233.
  • DeBlassie, R. Dante. Iterated Brownian motion in an open set. Ann. Appl. Probab. 14 (2004), no. 3, 1529–1558.
  • Csáki, E.; Földes, A. How small are the increments of the local time of a Wiener Ann. Probab. 14 (1986), no. 2, 533–546.
  • Csáki, E.; Csörgö, M.; Földes, A.; Révész, P.. The local time of iterated Brownian motion. J. Theoret. Probab. 9 (1996), no. 3, 717–743.
  • Donsker, M. D.; Varadhan, S. R. S. On laws of the iterated logarithm for local times. Comm. Pure Appl. Math. 30 (1977), no. 6, 707–753.
  • Griffin, Philip S. Laws of the iterated logarithm for symmetric stable processes. Z. Wahrsch. Verw. Gebiete 68 (1985), no. 3, 271–285.
  • Hu, Y.; Pierre-Loti-Viaud, D.; Shi, Z.. Laws of the iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 8 (1995), no. 2, 303–319.
  • Kasahara, Yuji. Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18 (1978), no. 2, 209–219.
  • Kesten, Harry. An iterated logarithm law for local time. Duke Math. J. 32 1965 447–456.
  • Khoshnevisan, Davar; Lewis, Thomas M. Chung's law of the iterated logarithm for iterated Brownian Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 3, 349–359.
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
  • Mogul'skii, A. A. Small deviations in the space of trajectories. (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755–765.
  • E. Nane, Iterated Brownian motion in parabola-shaped domains, Potential Analysis, 24 (2006), 105-123.
  • E. Nane, Iterated Brownian motion in bounded domains in $RR{R}^{n}$, Stochastic Processes and Their Applications, 116 (2006), 905-916.
  • E. Nane, Higher order PDE's and iterated processes, Submitted, math.PR/0508262.
  • P. Révész, Random walk in random and non-random environments, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. xiv+332 pp. ISBN: 981-02-0237-7.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1994. xii+560 pp. ISBN: 3-540-57622-3
  • Shi, Zh.; Yor, M. Integrability and lower limits of the local time of iterated Brownian Studia Sci. Math. Hungar. 33 (1997), no. 1-3, 279–298.
  • Spitzer, Frank. Principles of random walk. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp.
  • Taqqu, Murad S. A bibliographical guide to self-similar processes and long-range 137–162, Progr. Probab. Statist., 11, Birkhäuser Boston, Boston, MA, 1986.
  • Taylor, S. J. Sample path properties of a transient stable process. J. Math. Mech. 16 1967 1229–1246.
  • Xiao, Yimin. Local times and related properties of multidimensional iterated J. Theoret. Probab. 11 (1998), no. 2, 383–408.