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Chung-type law of the iterated logarithm for continuous time random walk

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Abstract

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.

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References

  1. Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P. Limit theorems for coupled continuous time random walks. The Annals of Probability, 32(1): 730–756 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P. Limit theorem for continuous time random walks with two time scales. Journal of Applied Probability, 41(2): 455–466 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berkowitz, B., Cortis, A., Dentz, M., Scher, H. Modeling non-Fickian transport in geological formatioms as a continuous time random walk. Reviews of Geophysics, 44, RG2003, doi:10.1029/2005RG000178 (2006)

    Article  Google Scholar 

  4. Darling, D.A. The influence of the maximum term in the addition of independent random variables. Transactions of the American Mathematical Society, 73: 95–107 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  5. Heyde, C.C. A contribution to the theory of large deviations for sums of independent random variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 7: 303–308 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  6. Meerschaert, M.M, Nane, E., Xiao, Y. Correlated continuous time random walks. Statistics & Probability Letters, 79: 1194–1202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Meerschaert, M.M, Scalas, E. Coupled continuous time random walks in finance. Physica A: Statistical Mechanics and its Applications, 370: 114–118 (2006)

    Article  MathSciNet  Google Scholar 

  8. Meerschaert, M.M., Scheffler, H.P. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley Interscience, New York, 2001

    MATH  Google Scholar 

  9. Meerschaert, M.M., Scheffler, H.P. Limit theorems for continuous time random walks with infinite mean waiting times. Journal of Applied Probability, 41(3): 623–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meerschaert, M.M., Scheffler, H.P. Triangular array limits for continuous time random walks. Stochastic Processes and their Applications, 118: 1606–1633 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Metzler, R., Klafter, J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and Theoretical, 37: 161–208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Scalas, E. Five years of continuous-time random walks in econophysics. In: A. Namatame (ed.), Proceedings of WEHIA, Kyoto, 2004

    Google Scholar 

  13. Seneta, E. Regularly Varying Functions. Lecture Notes in Mathematics, 508, Springer, Berlin, 1976

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Acknowledgements

The author wishes to express his deep gratitude to a referee for his/her valuable comments on an earlier version which improve the quality of this paper.

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Authors and Affiliations

  1. School of Economics, Hangzhou Dianzi University, Hangzhou, 310018, China

    Wen-sheng Wang

  2. College of Science, Hangzhou Normal University, Hangzhou, 310036, China

    Wen-sheng Wang

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  1. Wen-sheng Wang
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Correspondence to Wen-sheng Wang.

Additional information

Supported in part by the National Natural Science Foundation of China under Grant No. 11671115 and the Natural Science Foundation of Zhejiang Province under Grant No. LY14A010025.

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Wang, Ws. Chung-type law of the iterated logarithm for continuous time random walk. Acta Math. Appl. Sin. Engl. Ser. 33, 959–966 (2017). https://doi.org/10.1007/s10255-017-0711-0

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  • DOI: https://doi.org/10.1007/s10255-017-0711-0

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