Time-scales for Gaussian approximation and its breakdown under a hierarchy of periodic spatial heterogeneities

Bernoulli

Volumn 1, Issue 1-2, 1995, Pages 81-123

  Bhattacharya, Rabi N ^a   Götze, Friedrich ^b  

^a INDIANA UNIVERSITY   (United States)

^b BIELEFELD UNIVERSITY   (Germany)

Author keywords

Diffusion processes; Gaussian limits; Time scales

Indexed keywords

EID: 84972495950     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/bj/1186078363     Document Type: Article

References (38)

1. Anderson, M.P. (1979) Using models to simulate the movement of contaminants through ground water flow systems. CRC Critical Reviews in Environmental Control. 9, 97-156.
2. Aris, R. (1956) On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc. London Ser. A. 219, 186-203.
3. Aronson, D.G. (1967) Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc., 73, 890-896.
4. Bensoussan, A., Lions, J.L. and Papanicolaou, G.C. (1978) Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland.
5. Bhattacharya, R.N. (1982) On the functional central limit theorem and the law of iterated logarithm for Markov processes. Z. Wahrscheinlichkeitstheorie und Verw. Geb., 60, 185-201.
6. Bhattacharya, R.N. (1985) A central limit theorem for diffusions with periodic coefficients. Arm. Probab., 13, 385-396.
7. Bhattacharya, R.N. and Gupta, V.K. (1983) A theoretical explanation of solute dispersion in saturated porous media at the Darey scale. Water Resources Res., 19, 938-944.
8. Bhattacharya, R.N. and Gupta, V.K. (1984) On the Taylor-Aris theory of solute dispersion in periodic porous media. SIAM J. Appl. Math., 44, 33-39.
9. Bhattacharya, R.N. and Ramasubramanian, S. (1988) On the central limit theorem for diffusions with almost periodic coefficients. Sankhya Ser. A. 50, 9-25.
10. Bhattacharya, R.N. and Ranga Rao (1976) Normal Approximation and Asymptotic Expansions. New York: Wiley.
11. Bhatiacharya, R.N. and Waymire, E.C. (1990) Stochastic Processes with Applications. New York: Wiley.
12. Bhattacharya, R.N., Gupta, V.K., and Walker, H.F. (1989) Asymptotics of solute dispersion in periodic porous media. SIAM J. Appl. Math., 49, 86-98.
13. Billingsley, P. (1961) The Lindeberg-Levy theorem for martingales. Proc. Amer. Math. Soc., 12, 788-792.
14. Dagan, G. (1984) Solute transport in heterogeneous porous formations. J. Fluid Mech., 145, 151-177.
15. Diaconis, P. (1988) Group Representations in Probability and Statistics. IMS Lecture Notes. Hayward. CA: IMS.
16. Diaconis, P. and Stroock, D.W. (1991) Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab., 1, 36-61.
17. Einstein, A. (1905-8) Investigations on ¡he Theory of the Brownian Movement. New York: Dover.
18. Ethier, S.N. and Kurz, T.G. (1986) Markov Processes: Characterization and Convergence. New York: Wiley.
19. Fill, J.A. (1991) Eigenvalue bounds on convergence to stationarity for nonrevcrsiblc Markov chains, with an application to the exclusion process. Ann. Appl. Probah., 1, 62-87.
20. Fried, J.A. and Combarnous, M.A. (1971) Dispersion in porous media. Adv. Hydrosci., 7, 169-282.
21. Friedman, A. (1975) Stochastic Differential Equations and Applications, vol. 1. New York: Academic Press.
22. Gelhar, LAV. and Axness, C.L. (1983) Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resources Res., 19, 161-180.
23. Gihman, I.I., and Skorohod, A.V. (1989) Theory of Stochastic Processes, vol. 3. New York: Springer-Verlag.
24. Götze, F. and Hipp, C. (1983) Asymptotic expansions of sums of weakly dependent random vectors. Z. Wahrscheinlichkeitstheorie und Verw. Geh., 64, 211-239.
25. Götze, F. and Hipp, C. (1994) Asymptotic distributions of statistics in time series. Ann. Statist. 22, 2062-2088.
26. Gupta, V.K. and Bhattacharya, R.N. (1986) Solute dispersion in multidimensional periodic saturated porous media. Water Resources Res., 22, 156-164.
27. Guvcn, O. and Molz, F.J. (1986) Deterministic and stochastic analysis of dispersion in an unbounded stratified porous medium. Water Resources Res., 22, 1565-1574.
28. Hall, P.G. and Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. New York: Academic Press.
29. Kozlov, S.M. (1979) Averaging operators with almost periodic, rapidly oscillating coefficients. Math. USSR-Sb., 35, 481-498.
30. Kozlov, S.M. (1980) Averaging of random operators. Math. USSR-Sb., 37, 167-180.
31. Lahiri, S.N. (1993) Refinements in asymptotic expansions for sums of weakly dependent random vectors. Ann. Probab., 21, 791-799.
32. Lallemond-Barres, A. and Peaudecerf, P. (1978) Recherche des relations entre la valeur de la disperivité microscopique d’un milieu aquifere. des autres caractéristiques et les conditions de mesure: étude bibliographique. Bull. BRGM Ser. 2. Sect. III. 4, 277-284.
33. Molinary, J., Peaudecerf, P., Gaillard, B. and Launay, M. (1977) Essais conjoints en laboratoire et sur le terrain en vue. d’une approche simplifiée de la prévision des propagations des substances, miscibles dans les acquifers réels. Proc. Symp. on Hydrodynamic Diffusion and Dispersion in Porous Media. IAHR. Pavia. Italy, 89-102.
34. Papanicolaou, G.C. and Varadhan, S.R.S. (1979) Boundary problems with rapidly oscillating coefficients. Colloq. Math. Soc. Janos Bolyai. 27, 835-875.
35. Rogers, L.C.G. and Williams, D. (1987) Diffusions. Markov Processes, and Martingales, vol. 2. New York: Wiley.
36. Sposito, G., Jury, W.A. and Gupta, V.K. (1986) Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and fields. Water Resources Res., 22, 77-99.
37. Taylor, G.I. (1953) Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. London Ser. A. 219, 186-203.
38. Winter, C.L., Newman, C.M. and Neuman, S.P. (1984) A perturbation expansion for diffusion in a random velocity field. SIAM J. Appl. Math., 44, 425-442.