Research Article | Open Access
A. Chukwuemeka Okoroafor, "Some Local Asymptotic Laws for the Cauchy Process on the Line", International Journal of Stochastic Analysis, vol. 2007, Article ID 081934, 9 pages, 2007. https://doi.org/10.1155/2007/81934
Some Local Asymptotic Laws for the Cauchy Process on the Line
This paper investigates the lim inf behavior of the sojourn time process and the escape rate process associated with the Cauchy process on the line. The monotone functions associated with the lower asymptotic growth rate of the sojourn time are characterized and the asymptotic size of the large values of the escape rate process is developed.
- S. J. Taylor, “The measure theory of random fractals,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 3, pp. 383–406, 1986.
- Y. Xiao, “Random fractals and Markov processes,” in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot—Part 2, M. L. Lapidus and M. van Frankenhuijsen, Eds., vol. 72 of Proceedings of Symposia in Pure Mathematics, pp. 261–338, American Mathematical Society, Providence, RI, USA, 2004.
- S. J. Taylor and C. Tricot, “Packing measure, and its evaluation for a Brownian path,” Transactions of the American Mathematical Society, vol. 288, no. 2, pp. 679–699, 1985.
- A. C. Okoroafor and O. O. Ugbebor, “Lower asymptotic behaviour of the sojourn time for a stable process,” in Contemporary Stochastic Analysis (Ibadan, 1989), G. O. S. Ekhaguere, Ed., pp. 109–126, World Scientific, River Edge, NJ, USA, 1991.
- W. E. Pruitt and S. J. Taylor, “Packing and covering indices for a general Lévy process,” Annals of Probability, vol. 24, no. 2, pp. 971–986, 1996.
- B. Fristedt, “Sample functions of stochastic processes with stationary, independent increments,” in Advances in Probability and Related Topics, Vol. 3, pp. 241–396, Marcel Dekker, New York, NY, USA, 1974.
- E. A. Perkins and S. J. Taylor, “Uniform measure results for the image of subsets under Brownian motion,” Probability Theory and Related Fields, vol. 76, no. 3, pp. 257–289, 1987.
- S. Kochen and C. Stone, “A note on the Borel-Cantelli lemma,” Illinois Journal of Mathematics, vol. 8, pp. 248–251, 1964.
- D. Ray, “Some local properties of Markov processes,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif, 1965/1966), Vol. II: Contributions to Probability Theory—Part 2, pp. 201–212, University of California Press, Berkeley, Calif, USA, 1967.
- J. Takeuchi and S. Watanabe, “Spitzer's test for the Cauchy process on the line,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 3, pp. 204–210, 1964.
- A. C. Okoroafor and O. O. Ugbebor, “Upper rates of escape for stable process,” in Contemporary Stochastic Analysis (Ibadan, 1989), G. O. S. Ekhaguere, Ed., pp. 127–147, World Scientific, River Edge, NJ, USA, 1991.
Copyright © 2007 A. Chukwuemeka Okoroafor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.