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The lower envelope of positive self-similar Markov processes
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Loic Chaumont, Laboratoire de probabilités et modèles aléatoires
Juan Carlos Pardo Millan, Laboratoire de probabilités et modèles aléatoires
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Abstract
We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretzky and Erdös Math. Review 0047272 , Motoo Math. Review 0097866 , and Rivero Math. Review 2029617.
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Full text: PDF
Pages: 1321-1341
Published on: December 17, 2006
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Bibliography
- J. Bertoin. Lévy processes. Cambridge University Press, Cambridge, (1996) Math. Review 1406564
- J. Bertoin and M.E. Caballero. Entrance from $0+$ for increasing semi-stable Markov processes. Bernoulli, 8 (2002), no. 2,195--205, . Math. Review 1895890
- J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002), no. 4, 389--400. Math. Review 1918243
- N. Bingham, C.M. Goldie and J.L. Teugels. Regular variation. Cambridge University Press, Cambridge, 1989. Math. Review 1015093
- M.E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab., 34 (2006), no. 3, 1012--1034. Math. Review 2243877
- L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39--54. Math. Review 1419491
- L. Chaumont. Excursion normalisée, méandre et pont pour des processus stables. Bull. Sc. Math., 121 (1997), 377-403. Math. Review 1465814
- Y.S. Chow. On moments of ladder height variables. Adv. in Appl. Math. 7 (1986), no. 1, 46--54. Math. Review 0834219
- R.A. Doney. Stochastic bounds for Lévy processes. Ann. Probab., 32 (2004), no. 2, 1545--1552. Math. Review 2060308
- R.A. Doney and R.A. Maller. Stability of the overshoot for Lévy processes. Ann. Probab. 30 (2002), no. 1, 188--212. Math. Review 1894105
- A. Dvoretzky and P. Erdös. Some problems on random walk in space. Proceedings of the Second Berkeley Symposium. University of California Press, Berkeley and Los Angeles, 1951. Math. Review 0047272
- S. Janson. Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. Appl. Probab., 18 (1986), 865-879. Math. Review 0867090
- S. Kochen and C. Stone. A note on the Borel-Cantelli lemma. Illinois J. Math., 8 (1964), 248--251. Math. Review 0161355
- J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc., 104 (1962), 62--78. Math. Review 0138128
- J. Lamperti. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22 (1972), 205--225. Math. Review 0307358
- K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes, Stochastic Process. Appl., 116 (2006), 156--177. Math. Review 2197972
- M. Motoo. Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math., 10 (1958), 21--28. Math. Review 0097866
- V. Rivero. A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep., 75 (2003), no. 6, 443--472. Math. Review 2029617
- V. Rivero. Recurrent extensions of self-similar Markov processes and Cram'er's condition. Bernoulli, 11 (2005), no. 3, 471--509. Math. Review 2146891
- T. Watanabe. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields, 104 (1996), no. 3, 349--374. Math. Review 1376342
- Y. Xiao. Asymptotic results for self-similar Markov processes. Asymptotic methods in probabilty and statistics (Ottawa, ON, 1997), 323-340, North-Holland, Amsterdam, 1998. Math. Review 1661490
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Electronic Journal of Probability. ISSN: 1083-6489 |
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