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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.