The Annals of Probability

A functional LIL for symmetric stable processes

Xia Chen, James Kuelbs, and Wenbo Li

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A functional law of the iterated logarithm is obtained for symmetric stable processes with stationaryindependent increments.This extends the classical liminf results of Chung for Brownian motion, and of Taylor for such remaining processes. It also extends an earlier result of Wichura on Brownian motion.Proofs depend on small ball probability estimates and yield the small ball probabilities of the weighted sup-norm for these processes.

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Ann. Probab., Volume 28, Number 1 (2000), 258-276.

First available in Project Euclid: 18 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B17 60G17: Sample path properties 60J30

Functional LIL symmetric stable processes small ball probabilities


Chen, Xia; Kuelbs, James; Li, Wenbo. A functional LIL for symmetric stable processes. Ann. Probab. 28 (2000), no. 1, 258--276. doi:10.1214/aop/1019160119.

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  • de Acosta, A. (1983). Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab. 11 78-101.
  • Chung, K. L. (1948). On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. 64 205-233.
  • Cs´aki, E. (1980). A relation between Chung's and Strassen's law of the iterated logarithm.Wahrsch. Verw. Gebiete 54 287-301.
  • Donsker, M. D. and Varadhan, S. R. S. (1977). On laws of the iterated logarithm for local times. Comm. Pure Appl. Math. 30 707-753.
  • Kuelbs, J., Li, W. V. and Talagrand, M. (1994). Lim inf results for Gaussian samples and Chung's functional LIL. Ann. Probab. 22 1879-1903.
  • Li, W. V. (1998). Small deviations for Gaussian Markov processes under the sup-norm. J. Theoret. Probab. 12 971-984.
  • Marcus, M. B. and Pisier, G. (1984). Characterizations of almost surelycontinuous p-stable random Fourier series and stronglystationaryprocesses. Acta Math. 152 245-301.
  • Mogul'skii, A. A. (1974). Small deviations in a space of trajectories. Theoret. Probab. Appl. 19 726-736.
  • Mogul'skii, A. A. (1982). The Fourier method for finding the asymptotic behavior of small deviations of a Wiener process. Sibirsk. Mat. Zh. 23 161-174.
  • Samorodnitsky, G. (1998). Lower tails of self-similar stable processes. Bernoulli 4 127-142.
  • Taylor, S. J. (1967). Sample path properties of a transient stable process. J. Math. Mech. 16 1229-1246.
  • Wichura, M. (1973). A functional form of Chung's Law of the iterated logarithm for maximum absolute partial sums. Unpublished manuscript.