The Lower Envelope of Positive Self-Similar Markov Processes

Loic Chaumont (Laboratoire de probabilités et modèles aléatoires)
Juan Carlos Pardo Millan (Laboratoire de probabilités et modèles aléatoires)

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 Erdos (1951), Motoo (1958), and Rivero (2003).

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Pages: 1321-1341

Publication Date: December 17, 2006

DOI: 10.1214/EJP.v11-382

References

  1. J. Bertoin. LÈvy processes. Cambridge University Press, Cambridge, (1996) Math. Review 1406564
  2. 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
  3. J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of LÈvy processes. Potential Anal. 17 (2002), no. 4, 389--400. Math. Review 1918243
  4. N. Bingham, C.M. Goldie and J.L. Teugels. Regular variation. Cambridge University Press, Cambridge, 1989. Math. Review 1015093
  5. M.E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of LÈvy processes. Ann. Probab., 34 (2006), no. 3, 1012--1034. Math. Review 2243877
  6. L. Chaumont. Conditionings and path decompositions for LÈvy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39--54. Math. Review 1419491
  7. L. Chaumont. Excursion normalisÈe, mÈandre et pont pour des processus stables. Bull. Sc. Math., 121 (1997), 377-403. Math. Review 1465814
  8. Y.S. Chow. On moments of ladder height variables. Adv. in Appl. Math. 7 (1986), no. 1, 46--54. Math. Review 0834219
  9. R.A. Doney. Stochastic bounds for LÈvy processes. Ann. Probab., 32 (2004), no. 2, 1545--1552. Math. Review 2060308
  10. R.A. Doney and R.A. Maller. Stability of the overshoot for LÈvy processes. Ann. Probab. 30 (2002), no. 1, 188--212. Math. Review 1894105
  11. A. Dvoretzky and P. Erdˆs. 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
  12. 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
  13. S. Kochen and C. Stone. A note on the Borel-Cantelli lemma. Illinois J. Math., 8 (1964), 248--251. Math. Review 0161355
  14. J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc., 104 (1962), 62--78. Math. Review 0138128
  15. J. Lamperti. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22 (1972), 205--225. Math. Review 0307358
  16. K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of LÈvy processes, Stochastic Process. Appl., 116 (2006), 156--177. Math. Review 2197972
  17. M. Motoo. Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math., 10 (1958), 21--28. Math. Review 0097866
  18. 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
  19. V. Rivero. Recurrent extensions of self-similar Markov processes and Cram\'er's condition. Bernoulli, 11 (2005), no. 3, 471--509. Math. Review 2146891
  20. T. Watanabe. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields, 104 (1996), no. 3, 349--374. Math. Review 1376342
  21. 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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