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Random Fields with Pólya Correlation Structure

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Richard Finlay  and
Eugene Seneta
Richard Finlay*
Affiliation:
University of Sydney
Eugene Seneta*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
∗ Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
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Abstract

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We construct random fields with Pólya-type autocorrelation function and dampened Pólya cross-correlation function. The marginal distribution of the random fields may be taken as any infinitely divisible distribution with finite variance, and the random fields are fully characterized in terms of their joint characteristic function. This makes available a new class of non-Gaussian random fields with flexible correlation structure for use in modeling and estimation.

Keywords

Random fieldinfinitely divisible distributionPólya autocorrelation

MSC classification

Primary: 60G60: Random fields
Secondary: 60G10: Stationary processes 60E07: Infinitely divisible distributions; stable distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1037 - 1050
Copyright
© Applied Probability Trust 

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Christakos, G. (1984). On the problem of permissible covariance and variogram models. Water Resour. Res. 20, 251–265.CrossRefGoogle Scholar
Chung, K. L. (2001). A Course in Probability Theory, 3rd edn. Academic Press, San Diego, CA.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. John Wiley, New York.Google Scholar
Du, J. and Ma, C. (2013). Vector random fields with compactly supported covariance matrix functions. J. Statist. Planning Infer. 143, 457–467.CrossRefGoogle Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Finlay, R. and Seneta, E. (2007). A gamma activity time process with noninteger parameter and self-similar limit. J. Appl. Prob. 44, 950–959.CrossRefGoogle Scholar
Finlay, R., Fung, T. and Seneta, E. (2011). Autocorrelation functions. Internat. Statist. Rev. 79, 255–271.CrossRefGoogle Scholar
Gikhman, I. I. and Skorokhod, A. V. (1969). Introduction to the Theory of Random Processes. W. B. Saunders, Philadelphia, PA.Google Scholar
Khoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. Springer, New York.CrossRefGoogle Scholar
Kolmogoroff, A. (1923). Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue. Bull. Internat. Acad. Polon. Sci. Lett. Cl. Ser. Sci. Math. Nat. A, 1923, 83–86.Google Scholar
Lukacs, E. (1960). Characteristic Functions. Hafner, New York.Google Scholar
Ma, C. (2009). Construction of non-Gaussian random fields with any given correlation structure. J. Statist. Planning Inf. 139, 780–787.CrossRefGoogle Scholar
Ma, C. (2011a). Covariance matrices for second-order vector random fields in space and time. IEEE Trans. Signal Process. 59, 2160–2168.CrossRefGoogle Scholar
Ma, C. (2011b). Covariance matrix functions of vector χ2 random fields in space and time. IEEE Trans. Commun. 59, 2554–2561.CrossRefGoogle Scholar
Ma, C. (2011c). Vector random fields with long-range dependence. Fractals 19, 249–258.CrossRefGoogle Scholar
Ma, C. (2011d). Vector random fields with second-order moments or second-order increments. Stoch. Anal. Appl. 29, 197–215.CrossRefGoogle Scholar
Marfè, R. (2012). A multivariate pure-jump model with multi-factorial dependence structure. Int. J. Theoret. Appl. Finance 15, 1250028.CrossRefGoogle Scholar
Marfè, R. (2014). Multivariate Lévy processes with dependent Jump intensity. Quant. Finance 14, 1383–1398.CrossRefGoogle Scholar
Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin.CrossRefGoogle Scholar
Pólya, G. (1949). Remarks on characteristic functions. In Proc. Berkeley Symp. Math. Statist. Prob. 1945, 1946, University of California Press, Berkeley, CA, pp. 115–123.Google Scholar
Young, W. H. (1913). On the Fourier Series of bounded functions. Proc. London Math. Soc. 12, 41–70.CrossRefGoogle Scholar
Zygmund, A. (1968). Trigonometric Series, Vol. I, 2nd edn. Cambridge University Press.Google Scholar