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Limit theorems for continuous-time random walks in the double-array limit scheme

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References

  1. B. Baeumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, “Subordinated advection-dispersion equation for contaminant transport,” Water Resour. Res., 37, No. 6, 1543–1550 (2001).

    Article  Google Scholar 

  2. I. I. Banis, “Convergence rate estimates in the integral limit theorem,” Lit. Math. J., 12, No. 1, 41–46 (1972).

    MATH  MathSciNet  Google Scholar 

  3. G. M. Batanov, “Statistical properties and radial structure of plasma turbulence in the boundary region of the L2-M stellarator,” Plasma Phys. Control. Fusion, 40, 1241–1250 (1998).

    Article  Google Scholar 

  4. J. T. Bendler and M. F. Shlesinger, The Wonderful World of Stochastics, North-Holland, Amsterdam (1985).

    MATH  Google Scholar 

  5. V. Bening and V. Korolev, Generalized Poisson Models and Their Applications in Insurance and Finance, VSP, Utrecht, The Netherlands (2002).

    MATH  Google Scholar 

  6. D. A. Benson, M. M. Meerschaert, R. Schumer, and S. W. Wheatcraft, “Eulerian derivation of the fractional advection-dispersion equation,” J. Contaminant Hydrology, 38, 69–88 (2001).

    Google Scholar 

  7. D. A. Benson, M. M. Meerschaert, R. Schumer, and S. W. Wheatcraft, “Fractional dispersion, Lévy motion, and the MADE tracer tests,” Trans. Porous Media, 42, 211–240 (2001).

    Article  MathSciNet  Google Scholar 

  8. D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, “Application of a fractional advection-dispersion equation,” Water Resour. Res., 36, No. 6, 1403–1412 (2000).

    Article  Google Scholar 

  9. J.-P. Bouchaud and A. Georges, “Anomalous di.usion in disordered media: Statistical mechanics, models, and physical applications,” Phys. Rep., 145, 127–293 (1990).

    Article  MathSciNet  Google Scholar 

  10. H. Cartan, Calcul Différentiel. Formes Différentielles, Hermann, Paris (1967).

    Google Scholar 

  11. G. Christoph, “Uber notwendige und hinreichende Bedingungen fur Konvergenzaussagen im Falle einer stabilen Grenzverteilung,” Z. Wahrsch. verw. Geb., 54, No. 1, 29–40 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Christoph and W. Wolf, Convergence Theorems with a Stable Limit Law, Akademie Verlag, Berlin (1992).

    MATH  Google Scholar 

  13. M. B. Isichenko, “Percolation, statistical topography, and transport in random media,” Rev. Mod. Phys., 64, 961–1043 (1992).

    Article  MathSciNet  Google Scholar 

  14. B. V. Gnedenko and H. Fahim, “A certain transfer theorem,” Sov. Math. Dokl., 187, No. 1, 15–17 (1969).

    MATH  MathSciNet  Google Scholar 

  15. B. V. Gnedenko and V. Yu. Korolev, Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton, Fl (1996).

    Google Scholar 

  16. A. Karoblis, “On approximation to the distributions of sums of independent random variables,” Lit. Math. J., 23, No. 1, 101–107 (1983).

    MATH  MathSciNet  Google Scholar 

  17. J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A, 35, 3081 (1987).

    Article  MathSciNet  Google Scholar 

  18. J. Klafter, G. Zumofen, and M. F. Shlesinger, Physica A, 200, 222 (1993).

    Article  Google Scholar 

  19. V. Kolokoltsov, V. Korolev, and V. Uchaikin, Fractional Stable Distributions, Research Report No. 23/00, Nottingham Trent University (2000).

  20. V. Kolokoltsov, V. Korolev, and V. Uchaikin, “Fractional stable distributions,” J. Math. Sci., 105, No. 6, 2569–2576 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Kolokoltsov, V. Bening, V. Korolev, and T. Sukhorukova, Convergence Rate Estimates for Superpositions of Independent Stochastic Processes with Applications to Estimation of the Accuracy of Approximation of the Distributions of Continuous-Time Random Walks by Fractional Stable Laws, Preprint, The Nottingham Trent University (2003).

  22. V. Yu. Korolev, “A general theorem on the limit behavior of superpositions of independent random processes with applications to Cox processes,” J. Math. Sci., 81, No. 5, 2951–2956 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Kotulski, “Asymptotic distributions of the continuous-time random walks: A probabilistic approach,” J. Statist. Phys., 81, 777–779 (1995).

    Article  Google Scholar 

  24. V. M. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums [in Russian], Moscow State Univ. Press, Moscow (1990).

    MATH  Google Scholar 

  25. M. M. Meerschaert and H.-P. Scheffler, Limit Theorems for Continuous Time Random Walks, Preprint (http: //unr.edu/homepage/mcubed/LimitCTRW.pdf) (2001).

  26. M. M. Meerschaert and H.-P. Scheffler, “Stochastic solution of space-time fractional di.usion equations,” Phys. Rev. E, 65, No. 4, 1103–1106 (2002).

    Article  MathSciNet  Google Scholar 

  27. R. Metzler, E. Barkai, and J. Klafter, “Anomalous transport in disordered systems under the in.uence of external fields,” Physica A, 266, 343–350 (1999).

    Article  Google Scholar 

  28. A. A. Mitalauskas, “An estimate of convergence rate in the integral limit theorem in the case of convergence to a stable law,” Lit. Math. J., 11, No. 3, 627–639 (1971).

    MATH  MathSciNet  Google Scholar 

  29. E. W. Montroll and G. H. Weiss, “Random walks on lattices,” J. Math. Phys., 6, 167–181 (1965).

    Article  MathSciNet  Google Scholar 

  30. V. Paulauskas, “On a strengthening of the Lyapunov theorem,” Lit. Math. J., 9, No. 2, 323–328 (1969).

    MATH  MathSciNet  Google Scholar 

  31. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York (1994).

    MATH  Google Scholar 

  32. H. Scher and M. Lax, Phys. Rev. B, 7, 4491 (1973).

    Article  MathSciNet  Google Scholar 

  33. M. F. Shlesinger, B. J. West, and J. Klafter, Phys. Rev. Lett., 58, 1110 (1987).

    Article  MathSciNet  Google Scholar 

  34. M. F. Shlesinger, J. Statist. Phys., 36, 639 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  35. D. Szász, “On the rate of convergence in Lévy’s metric for random sums,” Progr. Statist., 2, 781–787 (1974).

    Google Scholar 

  36. V. V. Uchaikin and V. M. Zolotarev, Chance and Stability: Stable Distributions and Their Applications, VSP, Utrecht, The Netherlands (1999).

    Google Scholar 

  37. G. H. Weis and R. J. Rubin, Adv. Chem. Phys., 52, 363 (1983).

    Google Scholar 

  38. B. J. West and W. Deering, “Fractal physiology for physicists: Lévy statistics,” Phys. Rep., 246, 1–100 (1994).

    Article  Google Scholar 

  39. V. M. Zolotarev, “An estimate of the closeness of two convolutions of distributions,” in: International Vilnius Conference on Probability Theory and Mathematical Statistics. Abstacts of Communications, Vol. 1, Vilnius (1973), pp. 257–259.

    MathSciNet  Google Scholar 

  40. V. M. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence, RI (1986).

    MATH  Google Scholar 

  41. V. M. Zolotarev, Modern Theory of Summation of Random Variables, VSP, Utrecht (1997).

    MATH  Google Scholar 

  42. V. M. Zolotarev, “Natural estimates of convergence rate in the central limit theorem,” J. Math. Sci., 92, No. 4, 4112–4121 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  43. G. Zumofen, J. Klafter, and A. Blumen, Phys. Rev. E, 47, 2183 (1993).

    Article  Google Scholar 

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Supported by the Russian Foundation for Basic Research (grant Nos. 02-01-00949 and 02-01-01080) and by the Royal Society (grant No. gt/fSU/JP).

Proceedings of the Seminar on Stability Problems for Stochastic Models, Pamplona, Spain, 2003, Part III.

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Bening, V.E., Korolev, V.Y. & Kolokoltsov, V.N. Limit theorems for continuous-time random walks in the double-array limit scheme. J Math Sci 138, 5348–5365 (2006). https://doi.org/10.1007/s10958-006-0300-x

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