On the Law of the Iterated Logarithm for Local Times of Recurrent Random Walks

  • Xia Chen
Conference paper
Part of the Progress in Probability book series (PRPR, volume 47)


We consider the law of the iterated logarithm (LIL) for the local time of one-dimensional recurrent random walks. First we show that the constants in the LIL for the local time and for its supremum (with respect to the space variable) are equal under a very general condition given in Jain and Pruitt (1984). Second we evaluate the common value of the constants, as the random walk is in the domain of attraction of a not necessarily symmetric stable law. The first problem relies on a special maximal inequality established in this paper and the second on the LIL for Markovian additive functionals given in the author’s recent work.


Random Walk Local Time Iterate Logarithm Symmetric Stable Process Levy Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bertoin, J. (1995), Some applications of subordinators to local times of Markov processes, Forum Math. 7, 629–644.MathSciNetMATHCrossRefGoogle Scholar
  2. Chen, X. (1999), How often does a Harris recurrent Markov chain recur? Ann. Probab. 27, 1324–1346.MathSciNetMATHCrossRefGoogle Scholar
  3. Chung, K. L. and Hunt, C. A. (1949), On the zeros of , Ann. Math. 50, 385 - 400.MathSciNetMATHCrossRefGoogle Scholar
  4. Donsker, M. D. and Varadhan, S. R. S. (1977), On the law of the iterated logarithm for local times, Comm. Pure. Appl. Math. XXX, 707–753.MathSciNetCrossRefGoogle Scholar
  5. Feller, W. (1966), An Introduction to Probability Theory and Its applications, Vol. II, Wiley, New York.MATHGoogle Scholar
  6. Griffin, P. S., Jain, N. C. and Pruitt, W. E. (1984), Approximate local limit theorems for laws outside domains of attraction, Ann Probab. 12, 45–63.MathSciNetMATHCrossRefGoogle Scholar
  7. Jain, N. C. and Pruitt, W. E. (1984), Asymptotic behavior for the local time of a recurrent random walk, Ann. Probab. 12, 64–85.MathSciNetMATHCrossRefGoogle Scholar
  8. Kesten, H. (1965), An iterated logarithm law for local times, Duke Math. J. 32, 447–456.MathSciNetMATHCrossRefGoogle Scholar
  9. Le Gall, J.-F. and Rosen, J. (1991), The range of stable random walks, Ann. Probab. 19, 650–705MathSciNetMATHCrossRefGoogle Scholar
  10. Marcus, M. B. and Rosen, J. (1994), Law of the iterated logarithm for the local times of symmetric Lévy processes and recurrent random walks, Ann. Probab. 22, 626–658.MathSciNetMATHCrossRefGoogle Scholar
  11. Révész, P. (1990) Random Walk in Random and Non-Random Environments, World Scientific, London.MATHGoogle Scholar
  12. Revuz, D. (1975). Markov Chains, North-Holland, New York.MATHGoogle Scholar
  13. Zolotarev, V.M. (1986), One-dimensional Stable Distributions, Amer. Math. Soc., Providence, RI 1986.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Xia Chen
    • 1
  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

Personalised recommendations