Limit theorems and variation properties for fractional derivatives of the local time of a stable process

P. J. Fitzsimmons; R. K. Getoor

Annales de l'I.H.P. Probabilités et statistiques (1992)

  • Volume: 28, Issue: 2, page 311-333
  • ISSN: 0246-0203

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Fitzsimmons, P. J., and Getoor, R. K.. "Limit theorems and variation properties for fractional derivatives of the local time of a stable process." Annales de l'I.H.P. Probabilités et statistiques 28.2 (1992): 311-333. <http://eudml.org/doc/77434>.

@article{Fitzsimmons1992,
author = {Fitzsimmons, P. J., Getoor, R. K.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {occupation times; stable Markov processes; fractional derivatives; Hilbert transforms},
language = {eng},
number = {2},
pages = {311-333},
publisher = {Gauthier-Villars},
title = {Limit theorems and variation properties for fractional derivatives of the local time of a stable process},
url = {http://eudml.org/doc/77434},
volume = {28},
year = {1992},
}

TY - JOUR
AU - Fitzsimmons, P. J.
AU - Getoor, R. K.
TI - Limit theorems and variation properties for fractional derivatives of the local time of a stable process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 1992
PB - Gauthier-Villars
VL - 28
IS - 2
SP - 311
EP - 333
LA - eng
KW - occupation times; stable Markov processes; fractional derivatives; Hilbert transforms
UR - http://eudml.org/doc/77434
ER -

References

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  1. [B85] M.T. Barlow, Continuity of Local Times for Lévy Processes, Z. Wahrscheinlichkeitstheorie v. Geb., Vol. 69, 1985, pp. 23-35. Zbl0561.60076MR775850
  2. [Ba87] R.F. Bass, Lp Inequalities for Functionals of Brownian Motion, Sém. de Probabilités, XXI; Springer Lect. Notes Math., No. 1247, 1987, pp. 206-217. Zbl0616.60046MR941984
  3. [Be90] J. Bertoin, Complements on the Hilbert Transform and the Fractional Derivatives of Brownian Local Times, J. Math. Kyoto, Vol. 30, 1990, pp. 651-670. Zbl0725.60084MR1088348
  4. [BY87] Ph. Biane and M. Yor, Valeurs principales associées aux temps locaux browniens, Bull. Sc. Math., 2nd Ser., Vol. 111, 1987, pp. 23-101. Zbl0619.60072MR886959
  5. [Bi71] N.H. Bingham, Limit Theorems for Occupation Times of Markov Processes, Z. Wahrsheinlichkeitstheorie v. Geb., Vol. 17, 1971, pp. 1-22. Zbl0194.49503MR281255
  6. [BGT87] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987. Zbl0617.26001MR898871
  7. [BG68] R.M. Blumenthal and R.K. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968. Zbl0169.49204MR264757
  8. [Bo64] E.S. Boylan, Local Times for a Class of Markoff Processes, Illinois J. Math., Vol. 8, 1964, pp. 19-39. Zbl0126.33702MR158434
  9. [DK57] D.A. Darling and M. Kac, On Occupation Times for Markoff Processes, Trans. Am. Math. Soc., Vol. 84, 1957, pp.444-458. Zbl0078.32005MR84222
  10. [Da87] B. Davis, On the Barlow-Yor Inequalities for Local Time, Sém. de Probabilités, XXI; Springer Lect. Notes Math., Vol. 1247, 1987, pp. 218-220. Zbl0617.60041MR941985
  11. [Du91] R. Durrett, Probability: Theory and Examples, Wadsworth and Brooks/Cole, Pacific Grove, 1991. Zbl0709.60002MR1068527
  12. [F80] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980. Zbl0422.31007MR569058
  13. [FG91] P.J. Fitzsimmons and R.K. Getoor, On the Distribution of the Hilbert Transform of the Local Time of a Symmetric Lévy Process, Ann. Prob. (to appear). Zbl0767.60071MR1175273
  14. [G90] R.K. Getoor, Excessive Measures, Birkhäuser, Boston, 1990. Zbl0982.31500MR1093669
  15. [GK72] R.K. Getoor and H. Kesten, Continuity of Local Times for Markov Processes, Compositio Math., Vol. 24, 1872, pp. 277-303. Zbl0293.60069MR310977
  16. [HL28] G.H. Hardy and J.E. Littlewood, Some Properties of Fractional Integrals, I. Math. Zeit., Vol. 27, 1928, pp. 565-606. Zbl0003.15601MR1544927JFM54.0275.05
  17. [K77] Y. Kasahara, Limit Theorems of Occupation Times for Markov Processes. Publ. R.I.M.S. Kyoto Univ., Vol. 12, 1977, pp. 801-818. Zbl0367.60094MR448575
  18. [K81] Y. Kasahara, Two Limit Theorems for Occupation Times of Markov Processes, Jpn J. Math., Vol. 7, 1981, pp. 291-300. Zbl0483.60074MR729440
  19. [PY86] J. Pitman and M. Yor, Asymptotic Laws of Planar Brownian Motion, Ann. Prob., Vol. 14, 1986, pp. 773-779. Zbl0607.60070MR841582
  20. [R75] D. Revuz, Markov Chains, North-Holland, Amsterdam, 1975. Zbl0332.60045MR415773
  21. [S70] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. Zbl0207.13501MR290095
  22. [T48] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed., Clarendon Press, Oxford, 1948. 
  23. [T58] H.F. Trotter, A Property of Brownian Motion Paths, Illinois J. Math., Vol. 2, 1958, pp.425-433. Zbl0117.35502MR96311
  24. [Y85] T. Yamada, On the Fractional Derivative of Brownian Local Times, J. Math. Kyoto Univ., Vol. 25, 1985, pp. 49-58. Zbl0625.60090MR777245
  25. [Y86] T. Yamada, On Some Limit Theorems for Occupation Times of One Dimensional Brownian Motion and its Continuous Additive Functionals Locally of Zero Energy, J. Math. Kyoto Univ., Vol. 26, 1986, pp. 309-322. Zbl0618.60080MR849222

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