Bernoulli

  • Bernoulli
  • Volume 13, Number 4 (2007), 1053-1070.

Recurrent extensions of self-similar Markov processes and Cramér’s condition II

Víctor Rivero

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Abstract

We prove that a positive self-similar Markov process (X, ℙ) that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition.

Article information

Source
Bernoulli, Volume 13, Number 4 (2007), 1053-1070.

Dates
First available in Project Euclid: 9 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1194625602

Digital Object Identifier
doi:10.3150/07-BEJ6082

Mathematical Reviews number (MathSciNet)
MR2364226

Zentralblatt MATH identifier
1132.60056

Keywords
excursion theory exponential functionals of Lévy processes Lamperti’s transformation Lévy processes self-similar Markov processes

Citation

Rivero, Víctor. Recurrent extensions of self-similar Markov processes and Cramér’s condition II. Bernoulli 13 (2007), no. 4, 1053--1070. doi:10.3150/07-BEJ6082. https://projecteuclid.org/euclid.bj/1194625602


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References

  • Bertoin, J. and Caballero, M.-E. (2002). Entrance from $0+$ for increasing semi-stable Markov processes., Bernoulli 8 195--205.
  • Bertoin, J. and Doney, R.A. (1994). Cramér's estimate for Lévy processes., Statist. Probab. Lett. 21 363--365.
  • Bertoin, J. and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes., Potential Anal. 17 389--400.
  • Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes., Probab. Surv. 2 191--212.
  • Blumenthal, R.M. (1983). On construction of Markov processes., Z. Wahrsch. Verw. Gebiete 63 433--444.
  • Blumenthal, R.M. (1992)., Excursions of Markov Processes. Probability and Its Applications. Boston: Birkhäuser.
  • Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In, Exponential Functionals and Principal Values Related to Brownian Motion. Bibl. Rev. Mat. Iberoamericana 73--130. Madrid: Rev. Mat. Iberoamericana.
  • Dellacherie, C., Maisonneuve, B. and Meyer, P.-A. (1992)., Probabilités et potentiel: Processus de Markov (fin). Compléments du Calcul Stochastique. Paris: Hermann.
  • Erickson, K.B. (1970). Strong renewal theorems with infinite mean., Trans. Amer. Math. Soc. 151 263--291.
  • Fitzsimmons, P. (2006). On the existence of recurrent extensions of self-similar Markov processes., Electron. Comm. Probab. 11 230--241 (electronic).
  • Getoor, R.K. (1990)., Excessive Measures. Probability and Its Applications. Boston: Birkhäuser.
  • Getoor, R.K. and Sharpe, M.J. (1973). Last exit times and additive functionals., Ann. Probab. 1 550--569.
  • Goldie, C.M. (1991). Implicit renewal theory and tails of solutions of random equations., Ann. Appl. Probab. 1 126--166.
  • Lamperti, J. (1972). Semi-stable Markov processes. I., Z. Wahrsch. Verw. Gebiete 22 205--225.
  • Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes., Stochastic Process. Appl. 116 156--177.
  • Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér's condition., Bernoulli 11 471--509.
  • Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes., Ann. Probab. 22 546--565.