Large deviations for Riesz potentials of additive processes

Richard Bass; Xia Chen; Jay Rosen

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 626-666
  • ISSN: 0246-0203

Abstract

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We study functionals of the form ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|−σ ds1 ⋯ dsp, where X1(t), …, Xp(t) are i.i.d. d-dimensional symmetric stable processes of index 0<β≤2. We obtain results about the large deviations and laws of the iterated logarithm for ζt.

How to cite

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Bass, Richard, Chen, Xia, and Rosen, Jay. "Large deviations for Riesz potentials of additive processes." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 626-666. <http://eudml.org/doc/78037>.

@article{Bass2009,
abstract = {We study functionals of the form ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|−σ ds1 ⋯ dsp, where X1(t), …, Xp(t) are i.i.d. d-dimensional symmetric stable processes of index 0&lt;β≤2. We obtain results about the large deviations and laws of the iterated logarithm for ζt.},
author = {Bass, Richard, Chen, Xia, Rosen, Jay},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {large deviations; Riesz potentials; additive processes},
language = {eng},
number = {3},
pages = {626-666},
publisher = {Gauthier-Villars},
title = {Large deviations for Riesz potentials of additive processes},
url = {http://eudml.org/doc/78037},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Bass, Richard
AU - Chen, Xia
AU - Rosen, Jay
TI - Large deviations for Riesz potentials of additive processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 626
EP - 666
AB - We study functionals of the form ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|−σ ds1 ⋯ dsp, where X1(t), …, Xp(t) are i.i.d. d-dimensional symmetric stable processes of index 0&lt;β≤2. We obtain results about the large deviations and laws of the iterated logarithm for ζt.
LA - eng
KW - large deviations; Riesz potentials; additive processes
UR - http://eudml.org/doc/78037
ER -

References

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  1. [1] X. Chen. Large deviations and laws of the iterated logarithm for the local time of additive stable processes. Ann. Probab. 35 (2007) 602–648. Zbl1121.60025MR2308590
  2. [2] X. Chen, W. Li and J. Rosen. Large deviations for local times of stable processes and random walks in 1 dimension. Electron. J. Probab. 10 (2005) 577–608. Zbl1109.60016MR2147318
  3. [3] R. C. Dalang and J. B. Walsh. Geography of the level set of the Brownian sheet. Probab. Theory Related Fields 96 (1993) 153–176. Zbl0792.60038MR1227030
  4. [4] R. C. Dalang and J. B. Walsh. The structure of a Brownian bubble. Probab. Theory Related Fields 96 (1993) 475–501. Zbl0794.60047MR1234620
  5. [5] W. Donoghue. Distributions and Fourier Transforms. Academic Press, New York, 1969. Zbl0188.18102
  6. [6] M. Donsker and S. R. S. Varadhan. Asymtotics for the polaron. Comm. Pure Appl. Math. 36 (1983) 505–528. Zbl0538.60081MR709647
  7. [7] P.-J. Fitzsimmons and T. S. Salisbury. Capacity and energy for multiparameter stable processes. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 325–350. Zbl0689.60071MR1023955
  8. [8] F. Hirsch and S. Song. Symmetric Skorohod topology on n-variable functions and hierarchical Markov properties of n-parameter processes. Probab. Theory Related Fields 103 (1995) 25–43. Zbl0833.60074MR1347169
  9. [9] J.-P. Kahane. Some Random Series of Functions. Health and Raytheon Education Co., Lexington, MA, 1968. Zbl0192.53801MR254888
  10. [10] W. S. Kendall. Contours of Brownian processes with several-dimensional times. Z. Wahrsch. Verw. Gebiete 52 (1980) 267–276. Zbl0431.60056MR576887
  11. [11] D. Khoshnevisan. Brownian sheet images and Bessel–Riesz capacity. Trans. Amer. Math. Soc. 351 (1999) 2607–2622. Zbl0930.60055MR1638246
  12. [12] D. Khoshnevisan and Z. Shi. Brownian sheet and capacity. Ann. Probab. 27 (1999) 1135–1159. Zbl0962.60066MR1733143
  13. [13] D. Khoshnevisan and Y. Xiao. Level sets of additive Lévy processes. Ann. Probab. 30 (2002) 62–100. Zbl1019.60049MR1894101
  14. [14] V. Kolokoltsov. Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80 (2000) 725–768. Zbl1021.60011MR1744782
  15. [15] W. König and P. Mörters. Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002) 1605–1656. Zbl1032.60073MR1944002
  16. [16] J.-F. Le Gall, J. Rosen and N.-R. Shieh. Multiple points of Lévy processes. Ann. Probab. 17 (1989) 503–515. Zbl0684.60057MR985375
  17. [17] M. Marcus and J. Rosen. Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, NY, 2006. Zbl1129.60002MR2250510

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