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

Journal of Mathematical Sciences

Volumn 146, Issue 4, 2007, Pages 5959-5976

  Bening, V E ^a   Korolev, V Yu ^a   Koksharov, S ^a   Kolokoltsov, V N ^b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b NOTTINGHAM TRENT UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34848831455     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/s10958-007-0411-z     Document Type: Article

References (43)

1. B. Baeumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, "Subordinated advection-dispersion equation for contaminant transport," Water Resources Res., 37, No. 6, 1543-1550 (2001).
2. I. I. Banis, "Convergence rate estimates in the integral limit theorem," Lit. Math. J., 12, No. 1, 41-46 (1972).
3. G. M. Batanov, "Statistical properties and radial structure of plasma turbulence in the boundary region of the L2-M stellarator," Plasma Phys. Contr. Fusion, 40, 1241-1250 (1998).
4. J. T. Bendler and M. F. Shlesinger, in: M. F. Shlesinger and G. H. Weiss, Eds., The Wonderful World of Stochastics, North-Holland, Amsterdam (1985).
5. V. Bening and V. Korolev, Generalized Poisson Models and Their Applications in Insurance and Finance, VSP, Utrecht, The Netherlands (2002).
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).
7. D. A. Benson, M. M. Meerschaert, R. Schumer, and S. W. Wheatcraft, "Fractional dispersion, Lévy motion, and the MADE tracer tests," Transp. Porous Media, 42, 211-240 (2001).
8. D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, "Application of a fractional advection-dispersion equation," Water Resources Res., 36, No. 6, 1403-1412 (2000).
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).
10. H. Cartan, Calcul Différentiel. Formes Differentielles, Hermann, Paris (1967).
11. G. Christoph, "Über notwendige und hinreichende Bedingungen für Konvergenzaussagen im Falle einer stabilen Grenzverteilung," Z. Wahrsch. verw. Geb., 54, No. 1, 29-40 (1980).
12. G. Christoph and W. Wolf, Convergence Theorems with a Stable Limit Law, Akademie Verlag, Berlin (1992).
13. M. B. Isichenko, "Percolation, statistical topography, and transport in random media." Rev. Mod. Phys., 64, 961-1043 (1992).
14. B. V. Gnedenko and H. Fahim, "On a transfer theorem," Sov. Mat. Dokl. (1969).
15. B. V. Gnedenko and V. Yu. Korolev, Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton, Fl (1996).
16. A. Karoblis, "On approximation to the distributions of sums of independent random variables," Lit. Math. J., 23, No. 1, 101-107 (1983).
17. J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A, 35, 3081 (1987).
18. J. Klafter, G. Zumofen, and M. F. Shlesinger, Physica A, 200, 222 (1993).
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).
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).
23. M. Kotulskii, "Asymptotic distributions of the continuous-time random walks: A probabilistic approach," J. Statist. Phys., 81, 777-779 (1995).
24. V. M. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums [in Russian], Moscow State Univ. Press, Moscow (1990).
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 diffusion equations," Phys. Rev. E, 65, No. 4, 1103-1106 (2002).
27. R. Metzler, E. Barkai, and J. Klafter, "Anomalous transport in disordered systems under the influence of external fields," Physica A, 266, 343-350 (1999).
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).
29. E. W. Montroll and G. H. Weiss, J. Math. Phys., 6, 167 (1965).
30. V. Paulauskas, "On a strengthening of the Lyapunov theorem," Lit. Math. J., 9, No. 2, 323-328 (1969).
31. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York (1994).
32. H. Scher and M. Lax, Phys. Rev. B, 7, 4491 (1973).
33. M. F. Shlesinger, B. J. West, and J. Klafter, Phys. Rev. Lett., 58, 1110 (1987).
34. M. F. Shlesinger, J. Statist. Phys., 36, 639 (1984).
35. D. Szász, "On the rate of convergence in Lévy's metric for random indiced sums," in: Progr. Statist., Vol. 2, Amsterdam-London (1974), pp. 781-787.
36. V. V. Uchaikin and V. M. Zolotarev, Chance and Stability: Stable Distributions and Their Applications, VSP, Utrecht, The Netherlands (1999).
37. G. H. Weis and R. J. Rubin, Adv. Chem. Phys., 52, 363 (1983).
38. B. J. West and W. Deering, "Fractal physiology for physicists: Lévy statistics," Phys. Rep., 246, 1-100 (1994).
39. V. M. Zolotarev, "An estimate of the closeness of two convolutions of distributions," in: International Vilnius Conference on Probability Theory and Mathematical Statistics. Abstracts of Communications, Vol. 1, Vilnius (1973), pp. 257-259.
40. V. M. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence, RI (1986).
41. V. M. Zolotarev, Modern Theory of Summation of Random Variables, VSP, Utrecht (1997).
42. V. M. Zolotarev, "Natural estimates of convergence rate in the central limit theorem," J. Math. Sci., 92, No. 4, 4112-4121 (1998).
43. G. Zumofen, J. Klafter, and A. Blumen, Phys. Rev. E, 47, 2183 (1993).