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Laws of the iterated logarithm for α-time Brownian motion
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Erkan Nane, Purdue university |
Abstract
We introduce a class of iterated processes called
$alpha$-time Brownian motion for $0
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Full text: PDF
Pages: 434-459
Published on: June 19, 2006
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Bibliography
- Allouba, Hassan; Zheng, Weian. Brownian-time processes: the PDE connection and the half-derivative
Ann. Probab. 29 (2001), no. 4, 1780--1795. MR1880242 (2002j:60118)
- 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)
- 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 MR0898871 (88i:26004)
- Burdzy, Krzysztof. Some path properties of iterated Brownian motion.
67--87, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993. MR1278077 (95c:60075)
- Burdzy, Krzysztof; Khoshnevisan, Davar. The level sets of iterated Brownian motion.
231--236, Lecture Notes in Math., 1613, Springer, Berlin, 1995. MR1459464 (98k:60138)
- Chen, Xia; Li, Wenbo V. Small deviation estimates for some additive processes.
225--238, Progr. Probab., 55, Birkhäuser, Basel, 2003. MR2033891 (2004m:60115)
- 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. MR1761803 (2001e:60068)
- Chung, Kai Lai. On the maximum partial sums of sequences of independent random
Trans. Amer. Math. Soc. 64, (1948). 205--233. MR0026274 (10,132b)
- DeBlassie, R. Dante. Iterated Brownian motion in an open set.
Ann. Appl. Probab. 14 (2004), no. 3, 1529--1558. MR2071433 (2005f:60172)
- Csáki, E.; Földes, A. How small are the increments of the local time of a Wiener
Ann. Probab. 14 (1986), no. 2, 533--546. MR0832022 (87h:60147)
- Csáki, E.; Csörgö, M.; Földes, A.; Révész, P.. The local time of iterated Brownian motion.
J. Theoret. Probab. 9 (1996), no. 3, 717--743. MR1400596 (97f:60173)
- 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. MR0461682 (57 #1667)
- Griffin, Philip S. Laws of the iterated logarithm for symmetric stable processes.
Z. Wahrsch. Verw. Gebiete 68 (1985), no. 3, 271--285. MR0771467 (86d:60035)
- 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. MR1325853 (96b:60073)
- Kasahara, Yuji. Tauberian theorems of exponential type.
J. Math. Kyoto Univ. 18 (1978), no. 2, 209--219. MR0501841 (80g:40008)
- Kesten, Harry. An iterated logarithm law for local time.
Duke Math. J. 32 1965 447--456. MR0178494 (31 #2751)
- 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. MR1387394 (97k:60218)
- 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 MR1102015 (93c:60001)
- Mogulcprime skiui, A. A. Small deviations in the space of trajectories.
(Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755--765. MR0370701 (51 #6927)
- 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. R'{e}v'{e}sz, Random walk in random and non-random
environments, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. xiv+332 pp. ISBN: 981-02-0237-7.
MR1082348
- 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 MR1303781 (95h:60072)
- 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. MR1454115 (98i:60069)
- 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. MR0171290 (30 #1521)
- Taqqu, Murad S. A bibliographical guide to self-similar processes and long-range
137--162, Progr. Probab. Statist., 11, Birkhäuser Boston, Boston, MA, 1986. MR0899989 (88g:60091)
- Taylor, S. J. Sample path properties of a transient stable process.
J. Math. Mech. 16 1967 1229--1246. MR0208684 (34 #8493)
- Xiao, Yimin. Local times and related properties of multidimensional iterated
J. Theoret. Probab. 11 (1998), no. 2, 383--408. MR1622577 (99g:60136)
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Electronic Journal of Probability. ISSN: 1083-6489 |
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