Limiting behavior of a diffusion in an asymptotically stable environment

Arvind Singh

Annales de l'I.H.P. Probabilités et statistiques (2007)

  • Volume: 43, Issue: 1, page 101-138
  • ISSN: 0246-0203

How to cite


Singh, Arvind. "Limiting behavior of a diffusion in an asymptotically stable environment." Annales de l'I.H.P. Probabilités et statistiques 43.1 (2007): 101-138. <>.

author = {Singh, Arvind},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random environment; stable process; iterated logarithm law},
language = {eng},
number = {1},
pages = {101-138},
publisher = {Elsevier},
title = {Limiting behavior of a diffusion in an asymptotically stable environment},
url = {},
volume = {43},
year = {2007},

AU - Singh, Arvind
TI - Limiting behavior of a diffusion in an asymptotically stable environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2007
PB - Elsevier
VL - 43
IS - 1
SP - 101
EP - 138
LA - eng
KW - random environment; stable process; iterated logarithm law
UR -
ER -


  1. [1] J. Bertoin, Lévy processes, Cambridge Tracts in Math., vol. 121, Cambridge University Press, Cambridge, 1996. Zbl0861.60003MR1406564
  2. [2] J. Bertoin, On the first exit time of a completely asymmetric stable process from a finite interval, Bull. London Math. Soc.28 (5) (1996) 514-520. Zbl0863.60068MR1396154
  3. [3] J. Bertoin, R.A. Doney, On conditioning a random walk to stay nonnegative, Ann. Probab.22 (4) (1994) 2152-2167. Zbl0834.60079MR1331218
  4. [4] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia Math. Appl., vol. 27, Cambridge University Press, Cambridge, 1989. Zbl0667.26003MR1015093
  5. [5] A.A. Borovkov, Large deviations probabilities for random walks in the absence of finite expectations of jumps, Probab. Theory Related Fields125 (3) (2003) 421-446. Zbl1028.60021MR1967023
  6. [6] Th. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab.14 (4) (1986) 1206-1218. Zbl0608.60072MR866343
  7. [7] D. Cheliotis, One-dimensional diffusion in an asymmetric random environment, Ann. Inst. H. Poincaré Probab. Statist., in press, available at, Zbl1105.60077MR2269235
  8. [8] R.A. Doney, Conditional limit theorems for asymptotically stable random walks, Z. Wahrsch. Verw. Gebiete70 (3) (1985) 351-360. Zbl0573.60063MR803677
  9. [9] P. Erdös, On the law of the iterated logarithm, Ann. of Math. (2)43 (1942) 419-436. Zbl0063.01264MR6630
  10. [10] W. Feller, An Introduction to Probability Theory and its Applications, vol. II, John Wiley & Sons Inc., New York, 1966. Zbl0138.10207MR210154
  11. [11] C.C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable law, Sankhyā Ser. A30 (1968) 253-258. Zbl0182.22903MR240854
  12. [12] Y. Hu, Z. Shi, The limits of Sinai's simple random walk in random environment, Ann. Probab.26 (4) (1998) 1477-1521. Zbl0936.60088MR1675031
  13. [13] K. Itô, H.P. McKean, Diffusion Processes and their Sample Paths, Grundlehren Math. Wiss., Band 125, Academic Press Inc., New York, 1965. Zbl0285.60063MR199891
  14. [14] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren Math. Wiss., vol. 288, Springer-Verlag, Berlin, 1987. Zbl0635.60021MR959133
  15. [15] K. Kawazu, Y. Tamura, H. Tanaka, Localization of diffusion processes in one-dimensional random environment, J. Math. Soc. Japan44 (3) (1992) 515-550. Zbl0761.60072MR1167381
  16. [16] S. Kochen, Ch. Stone, A note on the Borel–Cantelli lemma, Illinois J. Math.8 (1964) 248-251. Zbl0139.35401
  17. [17] M.R. Pistorius, On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum, J. Theoret. Probab.17 (1) (2004) 183-220. Zbl1049.60042MR2054585
  18. [18] B.A. Rogozin, Distribution of the first ladder moment and height, and fluctuations of a random walk, Teor. Veroyatnost. i Primenen.16 (1971) 539-613. Zbl0269.60053MR290473
  19. [19] S. Schumacher, Diffusions with random coefficients, in: Particle Systems, Random Media and Large Deviations (Brunswick, Maine, 1984), Contemp. Math., vol. 41, Amer. Math. Soc., Providence, RI, 1985, pp. 351-356. Zbl0572.60053MR814724
  20. [20] Z. Shi, Sinai's walk via stochastic calculus, Survey paper, available at, MR2226845
  21. [21] A.V. Skorohod, Limit theorems for stochastic processes with independent increments, Teor. Veroyatnost. i Primenen.2 (1957) 145-177. Zbl0097.13001MR94842
  22. [22] V.M. Zolotarev, One-Dimensional Stable Distributions, Transl. Math. Monogr., vol. 65, Amer. Math. Soc., Providence, RI, 1986, (Translated from the Russian by H.H. McFaden, translation edited by Ben Silver). Zbl0589.60015MR854867

NotesEmbed ?


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.