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Power-law correlations and other models with long-range dependence on a lattice

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Chunsheng Ma
Chunsheng Ma*
Affiliation:
Wichita State University
*
∗Postal address: Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033, USA. Email address: cma@math.twsu.edu
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Abstract

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.

Keywords

Correlationlong-range dependencerandom fieldshort-range dependencespectral density

MSC classification

Primary: 60G60: Random fields
Secondary: 60G10: Stationary processes 62M30: Spatial processes 62M15: Spectral analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 690 - 703
Copyright
Copyright © Applied Probability Trust 2003 

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References

Berg, C., and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Springer, New York.Google Scholar
Besag, J. (1981). On a system of two-dimensional recurrence equations. J. R. Statist. Soc. B. 43, 302–309.Google Scholar
Brown, P. E., Diggle, P. J., Lord, M. E., and Young, P. C. (2001). Space-time calibration of radar rainfall data. Appl. Statist. 50, 221–241.Google Scholar
Cox, D. R. (1984). Long-range dependence: a review. In Statistics: an Appraisal, eds David, H. A. and David, H. T., Iowa State University Press, Ames, IA, pp. 55–74.Google Scholar
Cressie, N. A. C. (1993). Statistics for Spatial Data. John Wiley, New York.CrossRefGoogle Scholar
Donahue, M. J., Brockwell, P. J., and Davis, R. A. (1995). On permissible correlations for locally correlated stationary processes. Statist. Prob. Lett. 22, 49–53.Google Scholar
Fairfield Smith, H. (1938). An empirical law describing heterogeneity in the yields of agricultural crops. J. Agricultural Sci. 28, 1–23.CrossRefGoogle Scholar
Gneiting, T. (2000). Power-law correlations, related models for long-range dependence and their simulation. J. Appl. Prob. 37, 1104–1109.CrossRefGoogle Scholar
Granger, C. W. J., and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1, 15–29.Google Scholar
Haas, T. C. (1995). Local prediction of a spatio-temporal process with an application to wet sulfate deposition. J. Amer. Statist. Assoc. 90, 1189–1199.Google Scholar
Heyde, C. C., and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.CrossRefGoogle Scholar
Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Kashyap, R. L., and Lapsa, P. M. (1984). Synthesis and estimation of random fields using long-correlation models. IEEE Trans. Pattern Anal. Mach. Intell. 6, 800–809.Google Scholar
Ma, C. (2002). Correlation models with long-range dependence. J. Appl. Prob. 39, 370–382.Google Scholar
Ma, C. (2003a). Families of spatio-temporal stationary covariance models. To appear in J. Statist. Planning Infer.Google Scholar
Ma, C. (2003b). Spatio-temporal stationary covariance models. J. Multivariate Anal. 86, 97–107.CrossRefGoogle Scholar
Martin, R. J., and Eccleston, J. A. (1992). A new model for slowly-decaying correlations. Statist. Prob. Lett. 13, 139–145.Google Scholar
Martin, R. J., and Walker, A. M. (1997). A power-law model and other models for long-range dependence. J. Appl. Prob. 34, 657–670.Google Scholar
McLeod, A. I., and Hipel, K. W. (1978). Preservation of the rescaled adjusted range, 1: a reassessment of the Hurst phenomenon. Water Resources Res. 14, 491–508.Google Scholar
Modjeska, J. S., and Rawlings, J. O. (1983). Spatial correlation analysis of uniformity data. Biometrics 39, 373–384.CrossRefGoogle Scholar
Moran, P. A. P. (1973). Necessary conditions for Markovian processes on a lattice. J. Appl. Prob. 10, 605–612.Google Scholar
Pearce, S. C. (1976). An examination of Fairfield Smith's law of environmental variation. J. Agricultural Sci. 87, 21–24.CrossRefGoogle Scholar
Rainville, E. D. (1960). Special Functions. Macmillan, New York.Google Scholar
Renshaw, E. (1994). The linear spatial-temporal interaction process and its relation to 1/ω-noise. J. R. Statist. Soc. B. 56, 75–91.Google Scholar
Rodriguez-Iturbe, I., Marani, M., D'Odorico, P., and Rinaldo, A. (1998). On space-time scaling of cumulated rainfall fields. Water Resources Res. 34, 3461–3469.CrossRefGoogle Scholar
Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston, MA.Google Scholar
Whittle, P. (1956). On the variation of yield variance with plot size. Biometrika 43, 337–343.Google Scholar
Whittle, P. (1962). Topographic correlation, power-law covariance functions, and diffusion. Biometrika 49, 305–314.Google Scholar