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Moderate and Small Deviations for the Ranges of One-Dimensional Random Walks

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We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm.

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References

  1. Bass R. F., and Chen X. (2004). Self-intersection local time: critical exponent and laws of the iterated logarithm. Ann. Probab. 32, 3221-3247

    Article  MathSciNet  MATH  Google Scholar 

  2. Bass R. F., Chen X. and Rosen J. (2005). Large deviations for renormalized self-intersection local times of stable processes. Ann. Probab. 33, 984-1013

    Article  MathSciNet  MATH  Google Scholar 

  3. Bass, R. F., Chen, X., and Rosen, J. (2006). Moderate deviations for the range and self-intersections of planar random walks Memoirs of AMS (to appear).

  4. Bass R. F., and Kumagai T. (2002). Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30, 1369-1396

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen X. (2004). Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32, 3248-3300

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen X. (2005). Moderate deviations and law of the iterated logarithm for intersections of the ranges of random walks. Ann. Probab. 33, 1014-1059

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen X., and Li W. (2004). Large and moderate deviations for intersection local times. Probab Theor. Rel. Fields 128, 213-254

    Article  MATH  Google Scholar 

  8. Chen X., Li W., and Rosen J. (2005). Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 10, 577-608

    MathSciNet  Google Scholar 

  9. Chen X., and Rosen J. (2005). Exponential asymptotics for intersection local times of stable processes and random walks. Ann. l’Inst. H. Poincare 41, 908-928

    MathSciNet  Google Scholar 

  10. de Acosta A. (1980). Exponential moments of vector valued random series and triangular arrays. Ann. Probab. 8, 381-389

    MathSciNet  MATH  Google Scholar 

  11. de Acosta A. (1983). Small deviations in the functional central limit theorem with application to functional laws of the iterated logarithm. Ann. Probab. 11, 78-101

    MathSciNet  MATH  Google Scholar 

  12. Dembo A., and Zeitouni O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York

    MATH  Google Scholar 

  13. Dvoretzky, A., Erdös, P., and Kakutani, S. (1950). Double points of paths of Brownian motions in n-space. Leopoldo, Fejér et Frederico Riesz LXX annos natis dedicatus, Paris B, Acta Sci. Math. Szeged 12, 75–81.

    Google Scholar 

  14. Dvoretzky A., Erdöos P., and Kakutani S. (1954). Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3, 364-371

    MathSciNet  Google Scholar 

  15. Feller W. (1951). The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat. 22, 427-432

    MathSciNet  MATH  Google Scholar 

  16. Hamana Y. (1998). An almost sure invariance principle for the range of random walks. Stochastic Process. Appl. 78, 131-143

    Article  MathSciNet  MATH  Google Scholar 

  17. Jain N. C., and Pruitt W. E. (1971). The range of transient random walks. J. Anal. Math. 24, 369-393

    MathSciNet  MATH  Google Scholar 

  18. Jain N. C. and Pruitt W. E. (1972). The law of the iterated logarithm for the range of random walk. Ann. Math. Statist. 43, 1692-1697

    MathSciNet  MATH  Google Scholar 

  19. Jain N.C., and Pruitt W.E. (1974). Further limit theorems for the range of random walk. J. Anal. Math. 27, 94-117

    Article  MathSciNet  MATH  Google Scholar 

  20. Klenke A. and Mörters P. (2005).The multifractal spectrum of Brownian intersection local times. Ann. Probab. 33, 1255-1301

    Article  MathSciNet  MATH  Google Scholar 

  21. Le Gall J.-F. (1986a). Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection. Comm. Math. Phys. 104, 471-507

    Article  MathSciNet  MATH  Google Scholar 

  22. Le Gall J.-F. (1986b). Propriétés d’intersection des marches aléatoires. II. étude des cas critiques. Comm. Math. Phys. 104, 509-528

    Article  MathSciNet  MATH  Google Scholar 

  23. Le Gall J.-F. and Rosen J. (1991). The range of stable random walks. Ann. Probab. 19, 650-705

    MathSciNet  MATH  Google Scholar 

  24. Marcus M. B., and Rosen J. (1997) Laws of the iterated logarithm for intersections of random walks on Z 4. Ann. Inst. H. Poincaré Probab. Statist. 33, 37-63

    Article  MathSciNet  MATH  Google Scholar 

  25. Rosen J. (1997). Laws of the iterated logarithm for triple intersections of three-dimensional random walks. Electron. J. Probab. 2, 1-32

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Xia Chen

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  1. Xia Chen
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Correspondence to Xia Chen.

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We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm.

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Chen, X. Moderate and Small Deviations for the Ranges of One-Dimensional Random Walks. J Theor Probab 19, 721–739 (2006). https://doi.org/10.1007/s10959-006-0032-3

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  • DOI: https://doi.org/10.1007/s10959-006-0032-3

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